Dismal Arithmetic

David Applegate AT&T Shannon Labs 180 Park Avenue Florham Park, NJ 07932-0971 USA

                david@research.att.com

Marc LeBrun Fixpoint Inc. 448 Ignacio Blvd., #239 Novato, CA 94949 USA

                     mlb@well.com

N. J. A. Sloane^{1} AT&T Shannon Labs 180 Park Avenue Florham Park, NJ 07932-0971 USA

                 njas@research.att.com

To the memory of Martin Gardner (October 21, 1914 – May 22, 2010).

Abstract

Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number theory in this world, including analogues of the primes, number of divisors, sum of divisors, and the partition function.

1Corresponding author.

1 Introduction

To remedy the dismal state of arithmetic skills possessed by today’s children, we propose a “dismal arithmetic” that will be easier to learn than the usual version. It is easier because there are no carry digits and there is no need to add or multiply digits, or to do anything harder than comparing. In dismal arithmetic, for each pair of digits,

to Add, take the lArger, but to Multiply, take the sMaller.

That’s it! For example: 2 5 = 5, 2 5 = 2.

Addition or multiplication of larger numbers uses the same rules, always with the proviso that there are no carries. For example, the dismal sum of 169 and 248 is 269 and their dismal product is 12468 (Figure 1).

1 6 9
(2 4 8) / (2 6 9)

Fig. 1(a) Dismal addition.

1 6 9
2 4 8
1 6 8
1 4 4
1 2 2
1 2 4 6 8

Fig. 1(b) Dismal multiplication.

One might expect that nothing interesting could arise from such simple rules. However, developing the dismal analogue of ordinary elementary number theory will lead us to some surprisingly difficult questions.

Here are a few dismal analogues of standard sequences. The “even” numbers, 2 n, are

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 20, 21, 22, 22, … (1)

(entry \underline{A171818} in [17]). Note that n n (which is simply n) is a different sequence. For another, less obvious, analogue of the even numbers, see (13) in §3. The squares, n n, are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 200, … (2)

(\underline{A087019}),^{2} the dismal triangular numbers, 0 1 2 ⋯ n, are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 29, 29, 29, 29, 29, … (3)

(\underline{A087052}), and the dismal factorials, 1 2 ⋯ n, n ≥ 1, are

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110,

1111111110, 11111111110, 111111111100, 1111111111100, 11111111111100, …  (4)

(\underline{A189788}).

2See also the sums of two squares, \underline{A171120}. The numbers 10, 11, …, 99 are not the sum of any number of squares, so there is no dismal analogue of the four-squares theorem.

A formal definition of dismal arithmetic is given in §2, valid for any base b, not just base 10, and it shown there that the commutative, associative, and distributive laws hold (Theorem 1). In that section we also introduce the notion of a “digit map,” in order to study how changing individual digits in a dismal calculation affects the answer (Theorem 3, Corollary 4).

The dismal primes are the subject of §3. A necessary condition for a number to be a prime is that it contain a digit equal to b − 1. The data suggest that if k is large, almost all numbers of length k containing b−1 as a digit and not ending with zero are prime, and so the number of primes of length k appears to approach (b − 1)^{2} b^{k−2} as k → ∞ (Conjecture 10). In any case, any number with a digit equal to b − 1 is a product of primes (Theorem 7), and every number can be written as r times a product of primes, for some r ∈ {0, 1, …, b − 1} (Corollary 8). These factorizations are in general not unique. There is a useful process using digit maps for “promoting” a prime from a lower base to a higher base, which enables us to replace the list of all primes by a shorter list of prime “templates” (Table 3).

Dismal squares are briefly discussed in §4.

In §5 we investigate the different ways to order the dismal numbers, and in particular the partially ordered set defined by the divisibility relation (see Table 1). We will see that greatest common divisors and least common multiples need not exist, so this poset fails to be a lattice. On the other hand, we do have the notion of “relatively prime” and we can define an analogue of the Euler totient function.

In §6 we study the number-of-divisors function d_{b}(n), and investigate which numbers have the most divisors. It appears that in any base b ≥ 3, the number n = (b^{k} − 1)/(b − 1) = 111 … 1|_{b} has more divisors than any other number of length k. The binary case is slightly different. Here it appears that among all k-digit numbers n, the maximal value of d_{2}(n) occurs at n = 2^{k} − 2 = 111 … 10|_{2}, and this is the unique maximum for n ≠ 2, 4. Among all odd k-digit numbers n, d_{2}(n) has a unique maximum at n = 2^{k} − 1 = 111 … 111|_{2}, and if k ≥ 3 and k ≠ 5, the next largest value occurs at n = 2^{k} − 3 = 111 … 101|_{2}, its reversal 2^{k} − 2^{k−2} − 1 = 101 … 111|_{2}, and possibly other values of n (see Conjectures 12-14). Although we cannot prove these conjectures, we are able to determine the exact values of d_{b}(111 … 111|_{b}) and d_{2}(111 … 101|_{2}) (Theorem 17, which extends earlier work of Richard Schroeppel and the second author, and Theorem 18).

The sequence of the number of divisors of 11 … 11|_{2} (with k 1’s) turns out to arise in a variety of different problems, involving compositions, trees, polyominoes, Dyck paths, etc.—see Remark (iii) following Theorem 16. The initial terms can be seen in Table 8. This sequence appears in two entries in [17], \underline{A007059} and \underline{A079500}, and is the subject of a survey article by Frosini and Rinaldi [6]. The asymptotic behavior of this sequence was determined by Kemp [10] and by Knopfmacher and Robbins [12], the latter using the method of Mellin transforms—see (23). This is an example of an asymptotic expansion where the leading term has an oscillating component which, though small, does not go to zero. It is amusing to note that one of the first problems in which the asymptotic behavior was shown to involve a nonvanishing oscillating term was the analysis of the average number of carries when two k-digit numbers are added (Knuth [13], answering a question of von Neumann; see also Pippenger [18]). Here we see a similar phenomenon when there are no carries. In studying Conjectures 13 and 14, we observed that the numbers of divisors for the runners-up, 2^{k} − 3 and 2^{k} − 2^{k−2} − 1, appeared to be converging to one-fifth of the number of divisors of 11 … 11|_{2}. This is proved in Theorem 23. Our proof is modeled on Knopfmacher and Robbins’s proof [12] of (23), and we present the proof in such a way that it yields both results simultaneously.

The sum-of-divisors function σ_{b}(n) is the subject of §7. There are analogues of the perfect numbers, although they seem not to be as interesting as in the classical case. Section 8 discusses the dismal analogue of the partition function. Theorems 27 and 28 give explicit formulas for the number of partitions of n into distinct parts.

This is the second of a series of articles dealing with various kinds of carryless arithmetic, and contains a report of our investigations into dismal arithmetic carried out during the period 2000–2011. This work had its origin in a study by the second author into the results of performing binary arithmetic calculations with the usual addition and multiplication of binary digits replaced by other operations. If addition and multiplication are replaced by the logical operations OR and AND, respectively, we get base 2 dismal arithmetic. (If instead we use XOR and AND, the results are very different, the squares for example now forming the Moser-de Bruijn sequence \underline{A000695}.) Generalizing from base 2 to base 10 and then to an arbitrary base led to the present work.

In the first article in the series, [1], addition and multiplication were carried out “mod 10”, with no carries. A planned third part will discuss even more exotic arithmetics.

Although dismal arithmetic superficially resembles “tropical mathematics” [20], where addition and multiplication are defined by x ⊕ y := min{x, y}, x ⊙ y := x + y, there is no real connection, since tropical mathematics is defined over ℝ ∪ {∞}, uses carries, and is not base-dependent.

Notation

The base will be denoted by b and the largest digit in a base b expansion by β := b − 1. We write n = n_{k−1}n_{k−2} … n_{1}n_{0}|_{b} to denote the base b representation of the number ∑_{i=0}^{k−1} n_{i}b^{i}, and we define len_{b}(n) := k. The components n_{i} will be called the digits of n, even if b ≠ 10. In the examples in this paper b will be at most 10, so the notation n_{k−1} … n_{1}n_{0}|_{b} (without commas or spaces) is unambiguous. The symbols _{b} and _{b} denote dismal addition and multiplication, and we omit the base b if it is clear from the context. All non-bold operators (+, ×, <, etc.) refer to ordinary arithmetic operations, as do unqualified terms like “smallest,” “largest,” etc. We usually omit ordinary multiplication signs, but never dismal multiplication signs. We say that p divides n in base b (written p ≺_{b} n) if p _{b} q = n for some q, and that p = p_{k−1} … p_{1}p_{0}|_{b} is dominated by n = n_{k−1} … n_{1}n_{0}|_{b} (written p ≪_{b} n) if p_{i} ≤ n_{i} for all i. The symbol “|_{b}” always marks the end of a base b expansion of a number, and is never used for “divides in base b.”

2 Basic definitions and properties

We began, as we all did, in base 10, but from now on we will allow the base b to be an arbitrary integer ≥ 2.

Let A denote the set of base b “digits” {0, 1, 2, …, b − 1}, equipped with the two binary operations

m _{b} n := max{m, n}, m _{b} n := min{m, n}, for m, n ∈ A .  (5)

A dismal number is an element of the semiring A[X] of polynomials ∑_{i=0}^{k−1} n_{i}X^{i}, n_{i} ∈ A. If M[X] := ∑_{i=0}^{k−1} m_{i}X^{i} and N[X] := ∑_{i=0}^{l−1} n_{i}X^{i} are dismal numbers then their dismal sum is formed by taking the dismal sum of corresponding pairs of digits, analogously to ordinary addition of polynomials:

M[X] _{b} N[X] := ∑_{i=0}^{max{k,l}−1} p_{i}X^{i} ,  (6)

where p_{i} := m_{i} _{b} n_{i}, and their dismal product is similarly formed by using dismal arithmetic to convolve the digits, analogously to ordinary multiplication of polynomials:

M[X] _{b} N[X] := ∑_{i=0}^{k+l−2} q_{i}X^{i} ,  (7)

where

q_{0} := m_{0} _{b} n_{0} ,
q_{1} := (m_{0} _{b} n_{1}) _{b} (m_{1} _{b} n_{0}) ,
q_{2} := (m_{0} _{b} n_{2}) _{b} (m_{1} _{b} n_{1}) _{b} (m_{2} _{b} n_{0}) ,
⋯

We will identify a dismal number N(X) = ∑_{i=0}^{k−1} n_{i}X^{i} with the integer n whose base b expansion is n = ∑_{i=0}^{k−1} n_{i}b^{i} (n is obtained by evaluating the polynomial N(X) at X = b), and we define len_{b}(n) := k. The rules (5)-(7) then translate into the rules for dismal addition and multiplication stated in §1: there are no carries, and digits are combined according to the rules in (5). The b-ary number ∑_{i=0}^{k−1} n_{i}b^{i} will also be written as n_{k−1}n_{k−2} … n_{1}n_{0}|_{b}. Dismal numbers are, by definition, always identified with nonnegative integers. Note that len_{b}(m _{b} n) = max{len_{b}(m), len_{b}(n)} and len_{b}(m _{b} n) = len_{b}(m) + len_{b}(n) − 1.

Theorem 1. The dismal operations _{b} and _{b} on A[X] satisfy the commutative and associative laws, and _{b} distributes over _{b}.

Proof. (Sketch.) Each law requires us to show the identity of two polynomials, and so reduces to showing that certain identities hold for the coefficients of each individual degree in the two polynomials. These identities are assertions about min and max in the set A, which hold since (A, ≤) is a totally ordered set, and (A, min, max) is a distributive lattice (cf. [9]).

If R denotes the operation of reversing the order of digits and m and n have the same length, then R(m _{b} n) = R(m) _{b} R(n) and R(m _{b} n) = R(m) _{b} R(n).

Individual digits in a dismal sum or product can often be varied without affecting the result, so dismal subtraction and division will not be defined. Example: 16 _{10} 75 = 26 _{10} 75 = 76, 16 _{10} 75 = 16 _{10} 85 = 165. This is why dismal numbers form only a semiring. On the other hand, this semiring does have a multiplicative identity (see the next section), and there are no zero divisors.

In certain situations we can give a more precise statement about how changing digits in a dismal sum or product affects the result. We begin with a lemma about ordinary functions of real variables.

Lemma 2. Let f be a single-valued function of real variables x_{1}, …, x_{k}, k ≥ 2, formed by repeatedly composing the functions (x, y) ↦ min{x, y} and (x, y) ↦ max{x, y}. If g is a nondecreasing function of x, meaning that

x ≤ y ⇒ g(x) ≤ g(y) ,  (8)

then

f(g(x_{1}), …, g(x_{k})) = g(f(x_{1}, …, x_{k})) ,  (9)

for all real x_{1}, …, x_{k}.

We omit the easy inductive proof.

We define a base b digit map to be a nondecreasing function g mapping {0, 1, …, b−1} into itself. The map g need not be one-to-one or onto. If g is a digit map and n = n_{k−1} … n_{1}n_{0}|_{b} then we set g(n) := g(n_{k−1}) … g(n_{1})g(n_{0})|_{b}.

Theorem 3. If m and n are dismal numbers and g is a base b digit map, then

g(m _{b} n) = g(m) _{b} g(n) ,
g(m _{b} n) = g(m) _{b} g(n) .  (10)

Proof. This follows from Lemma 2, since the individual digits of m _{b} n and m _{b} n are functions of the digits of m and n of the type considered in that lemma.

Corollary 4. If p = m _{b} n then p can also be written as m′ _{b} n′, where m′ and n′ use only digits that are digits of p.

Proof. (Sketch.) Arrange all distinct digits occurring in p, m, and n in increasing order. Then construct a digit map g by increasing or decreasing the digits in m and n that are not in p until they coincide with digits of p, leaving the digits of p fixed.

For example, consider the product 165 = 16 _{10} 85 mentioned above. The digits involved are 1, 5, 6, 8, and the digit map described in the proof fixes 1, 5, and 6 and maps 8 to 6. The resulting factorization is 165 = 16 _{10} 65. (The additive analogue of Corollary 4 is true, but trivial.)

We will see other applications of Theorem 3 in the next section.

Note that when we are computing the base b dismal sum or product of two numbers p and q, once we have expressed p and q in base b, the value of b plays no further role in the calculation. Of course we need to know b when we convert the result back to an integer, but otherwise b is not used. So we have:

Lemma 5. If the largest digit that is mentioned in a base b dismal sum or product is d (where 0 ≤ d ≤ b − 1), then the same calculation is valid in any base that exceeds d.

For example, here is the calculation of the base 2 dismal product of 13 = 1101|_{2} and 5 = 101|_{2}:

1 1 0 1 _{2} 1 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1

This tells us that 13 _{2} 5 = 61, but the same tableau can be read in base 3, giving 37 _{3} 10 = 361, or in base 10, giving 1101 _{10} 101 = 111101.

Recall that we say that p divides n in base b (written p ≺_{b} n) if p _{b} q = n for some q. Since len_{b}(p) ≤ len_{b}(n), nonzero numbers have only finitely many divisors. We also say that p := p_{k−1} … p_{1}p_{0}|_{b} is dominated by n := n_{k−1} … n_{1}n_{0}|_{b} (written p ≪_{b} n) if p_{i} ≤ n_{i} for all i. Then p ≪_{b} n if and only if p _{b} n = n. Another consequence of Lemma 2 is:

Lemma 6. If p ≪_{b} m and q ≪_{b} n then p _{b} q ≪_{b} m _{b} n and p _{b} q ≪_{b} m _{b} n.

Finally, we remark without giving any details that, in any base, the sets of numbers with digits in nondecreasing order, or in nonincreasing order (see \underline{A009994} and \underline{A009996} for base 10) are closed under dismal addition and multiplication.

3 Dismal primes

In dismal arithmetic in base b, for bases b > 2, the multiplicative identity is no longer 1 (for example, 1 _{10} 23 = 11, not 23). In fact, it follows from the definition of multiplication that the multiplicative identity is the largest single-digit base b number, β := b − 1. For base b = 10 we have β = 9, and indeed the reader will easily check that 9 _{10} n = n for all n. An empty dismal product is defined to be β, by convention.

If p _{b} q = β, then p = q = β, so β is the only unit. We therefore define a prime in base b dismal arithmetic to be a number, different from β, whose only factorization is β times itself.

If p is prime, then at least one digit of p must equal β (for if the largest digit were r < β, then p = r _{b} p, and r would be a divisor of p). The base b expansions of the first few primes are 1β (this is the smallest prime), 2β, 3β … β − 1 β, β0, β1, …, ββ, 10β, . . .. In base 10, the primes are

19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, 219,

309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, … (11)

(\underline{A087097}). Notice that the presence of a digit equal to β is a necessary but not sufficient condition for a number to be a prime: 11β|_{b} = 1β|_{b} _{b} 1β|_{b} is not prime (see \underline{A087984} for these exceptions in the case b = 10). In base 2, the primes (written in base 2) are

10, 11, 101, 1001, 1011, 1101, 10001, 10011, 10111, 11001, 11101, 100001, …  (12)

(\underline{A171000}). In view of the interpretation of base 2 dismal arithmetic in terms of Boolean operations mentioned in §1, the corresponding polynomials

X, X + 1, X^{2} + 1, X^{3} + 1, X^{3} + X + 1, X^{3} + X^{2} + 1, X^{4} + 1, X^{4} + X + 1, …,

together with 1, might be called the OR-irreducible Boolean polynomials. Their decimal equivalents,

1, 2, 3, 5, 9, 11, 13, 17, 19, 23, 25, 29, 33, 35, 37, …

form sequence \underline{A067139} in [17], contributed by Jens Voß in 2002.

All numbers of the form 100 … 00β|_{b} (with zero or more internal zeros) are base b primes, since there is no way that p _{b} q can have the form 100 … 00β|_{b} unless p or q is a single-digit number. So there are certainly infinitely many primes in any base.

Since 1β|_{b} is the smallest prime, the numbers 1β|_{b} _{b} n are another analogue of the even numbers. Whereas the first version of the even numbers, given in (1) for base 10, was simply “replace all digits in n that are bigger than 2 with 2’s,” this version is more interesting. In base 10 we get

0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 111, 112, 113, 114, 115, 116, 117, 118, … (13)

(\underline{A162672}, which contains repetitions and is not monotonic).

If b > 2, there are numbers which cannot be written as a product of primes (e.g., 1).

Theorem 7. Any base b number with a digit equal to β is a (possibly empty) dismal product of dismal primes.

Proof. The number β itself is the empty product of primes. Every two-digit number with β as a digit is already a prime. If there are more than two digits, either the number is a prime, or it factorizes into the product of two numbers, both of which must have β as a digit. The result follows by induction.

Corollary 8. Every base b number can be written as r times a dismal product of dismal primes, for some r ∈ {0, 1, 2, …, b − 1}.

Proof. Let r be the largest digit of n. If r = β the result follows from the theorem. Otherwise, let m be obtained by changing all occurrences of r in the b-ary expansion of n to β’s, so that n = r _{b} m, and apply the theorem to m.

Even when it exists, the factorization into a dismal product of dismal primes is in general not unique. In base 10, for example, the list of numbers with at least two different factorizations into a product of primes is

1119, 1129, 1139, 1149, 1159, 1169, 1179, 1189, 1191, 1192, 1193, 1194, 1195, …, (14)

where for instance 1119 = 19 _{10} 19 _{10} 19 = 19 _{10} 109 (\underline{A171004}).

We can, of course, study the primes dividing n, even if n does not contain a digit equal to β. Without giving any details, we mention that [17] contains the following sequences: the number of distinct prime divisors of n (\underline{A088469}), their dismal sum (\underline{A088470}), and dismal

k base 2 base 10 k base 2 k base 2 k  base 2
1  0  0  11 323 21 442313 31 510471015
2  2  18  12 682 22 902921 32 1027067090
3  1  81  13 1424 23 1833029 33 2065390101
4  3  1539  14 2902 24 3719745 34 4151081457
5  5  20457 15 5956 25 7548521 35 8336751732
6  9  242217 16 12368 26 15264350 36 16734781946
7 19  2894799 17 25329 27 30859444 37 33583213577
8 39 33535839 18 51866 28 62355854 38 67357328359
9 77 381591711 19 106427 29 125773168 39 135056786787
10 168  ?  20 217216 30 253461052 40  ?

Table 1: Numbers of dismal primes with k digits in bases 2 and 10 (\underline{A169912}, \underline{A087636}).

product (\underline{A088471});^{3} also the lists of numbers n such that the dismal sum of the distinct prime divisors of n is < n (\underline{A088472}), ≤ n (\underline{A088473}), ≥ n (\underline{A088475}), > n (\underline{A088476}); as well as the numbers n such that the dismal product of the distinct prime divisors of n is < n (\underline{A088477}), ≤ n (\underline{A088478}), = n (\underline{A088479}), ≥ n (\underline{A088480}), and > n (\underline{A088481}). There is no analogue of \underline{A088476} or \underline{A088481} in ordinary arithmetic.

One omission from the above list is explained by the following theorem.

Theorem 9. In base b dismal arithmetic, n is prime if and only if the dismal sum of its distinct dismal prime divisors is equal to n.

Proof. If n = p is prime then the sum of the primes dividing it is p. Suppose n is not prime and let m be the sum of the distinct dismal primes diving n. If n is divisible by a prime p with len_{b}(p) = len_{b}(n), then n = r _{b} p, r < β, the largest digit in n is r, and so m ≠ n since n ≪_{b} p ≪_{b} m. If len_{b}(p) < len_{b}(n) for all prime divisors p, then len_{b}(m) < len_{b}(n), and again m ≠ n.

We now consider how many primes there are. Let π_{b}(k) denote the number of base b dismal primes with k digits. Table 1 shows the initial values of π_{2}(k) and π_{10}(k). Necessary conditions for a number n to be prime are that it contain β as a digit and (if k > 2) does not end with 0. There are

(b − 1)^{2} b^{k−2} − (b − 2) (b − 1)^{k−2}  (15)

such numbers. It seems likely that, as k increases, almost all of these numbers will be prime, and the data in Table 1 is consistent with this. We therefore make the following conjecture.

Conjecture 10.

π_{k}(b) ∼ (b − 1)^{2} b^{k−2} as k → ∞ .  (16)

3\underline{A088471} has an unusual beginning: 9, 9, 9, 9, 9, 9, 9, 9, 9, 90, 123456789987654321, 19, 19, 19, ….

We can get a lower bound on π_{b}(k) by producing large numbers of primes, using the process of “promotion.” We call a base b number with at least two digits a pseudoprime if its only factorizations are of the form n = p _{b} q where at least one of p and q has length 1. In base b, n is a prime if and only if it is a pseudoprime and contains a digit β. If n_{k−1}n_{k−2}...n_{0}|_{b} is a pseudoprime and r is its maximal digit, then n_{k−1}n_{k−2}...n_{0}|_{r+1} is a base r + 1 prime and furthermore n_{k−1}n_{k−2}...n_{0}|_{c} is a pseudoprime in any base c ≥ r + 1. In base 2 there is no difference between primes and pseudoprimes. As long as we exclude numbers ending with 0, reversing the digits of a number does not change its status as a prime or pseudoprime.

The advantage of working with pseudoprimes rather than primes is that the inverse image of a pseudoprime under a digit map (see §2) is again a pseudoprime.

Theorem 11. For a base b digit map g, if g(n_{k−1}n_{k−2}...n_{0}|_{b}) is a pseudoprime and g(n_{k−1}) is not 0, then n_{k−1}n_{k−2}...n_{0}|_{b} is a pseudoprime.

Proof. This follows immediately from Theorem 3.

So if p is a pseudoprime, then any number n with the property that there is a digit map sending n to p is also a pseudoprime; we think of n as being obtained by “promoting” p, and call p the “template” for n.

Here is an equivalent way to describe the promotion process. Suppose the distinct digits in p, the template, or number to be promoted, are d_{1} < d_{2} < d_{3} < · · ·. For each d_{i}, choose a set of digits S(d_{i}) such that all the digits in S(d_{i}) are strictly less than all those in S(d_{j}), for i < j. Replace any digit d_{i} in p by any digit in S(d_{i}). All numbers n obtained in this way are promoted versions of p (the required digit map g being defined by g(c) = d_{i} for all c ∈ S(d_{i})).

Any pseudoprime (in any base) with at most four digits can be obtained by promoting a base 2 prime. At length 2, 11|_{2} is a prime (see (12)), so every 2-digit number rs|_{b} is a pseudoprime, using the digit map g that sends r and s to 1. This is valid even if s = 0, since (8) still holds. At length 3, 101|_{2} is a base 2 prime, so any three-digit number rst|_{b} with r > s, t > s is a pseudoprime (take g(r) = g(t) = 1, g(s) = 0), and this captures all three-digit pseudoprimes. There are three templates of length 4, 1001|_{2}, 1011|_{2}, and 1101|_{2}. These can be promoted to capture all four-digit primes, which are the numbers rstu|_{b} for which one of the following holds:

r and u are both strictly greater than s and t , r, s, and u are strictly greater than t , r, t, and u are strictly greater than s .

For lengths greater than four, we must use some nonbinary templates to capture all pseudoprimes, and as the length k increases so does the fraction of nonbinary templates required, as shown in Table 2. The columns labeled (a) and (b) give the number of binary templates and the total number of templates, respectively, and the columns (c) and (d) give the number of base 10 primes obtained by promoting the templates in columns (a) and (b).

We can reduce the list of templates by omitting those that are reversals of others. Table 3 shows the reduced list of templates of lengths ≤ 6.

\begin{matrix}k & (a) & (b) & (c) & (d) \\ 2 & 1 & 1 & 18 & 18 \\ 3 & 1 & 1 & 81 & 81 \\ 4 & 3 & 3 & 1539 & 1539 \\ 5 & 5 & 8 & 17661 & 20457 \\ 6 & 9 & 51 & 135489 & 242217 \end{matrix}

Table 2: For lengths k = 2 through 6, the numbers of (a) binary templates, (b) all templates,

(c) base 10 primes obtained by promoting the binary templates, and (d) base 10 primes obtained by promoting all the templates.

11 100001 102212 120212
101 100011 102221 120221
1001 100101 103223 120222
1011 100111 103233 121022
10001 101011 110212 121102
10011 101221 112021 122102
10111 101222 112022 122202
12021 102201 120021 132023
12022 102202 120022 133023

Table 3: Reduced list of prime templates: every pseudoprime of length ≤ 6 can be obtained by promoting one of these 36 pseudoprimes or its reversal (\underline{A191420}).

Since 1 \underbrace{00 … 0}_{k−2} 1|_{2} is prime, the promotion process tells us for example that the numbers

r \underbrace{s_{1}s_{2} … s_{k−2}}_{k−2} β|_{b} are prime, for 1 ≤ r ≤ β, provided each of the digits s_{i} is in the range 0

through r − 1. This gives (b − 1)^{k−2} + 2(b − 2)^{k−2} + ⋯ = O(b^{k−2}) primes of length k, which for b > 2 is of exponential growth but smaller than (16).

4 Dismal squares

One might expect that it would be easier to find the number of dismal squares of a given length than the number of dismal primes, but we have not investigated squares as thoroughly, and we do not even have a precise conjecture about their asymptotic behavior. In base 2, the first few dismal squares, written in base 10, are

0, 1, 4, 7, 16, 21, 28, 31, 64, 73, 84, 95, 112, 125, 124, 127, 256, 273, …,  (17)

(\underline{A067398}, also contributed by Jens Voß in 2002), and the numbers of squares of lengths 1 (including 0), 3, 5, 7, … are

2, 2, 4, 8, 15, 29, 55, 105, 197, 367, 678, 1261, 2326, 4293, 7902, 14431, … (18)

(\underline{A190820}). In base 10, the first few squares were given in (2), and the numbers of squares of lengths 1 (including 0), 3, 5, 7, … are

10, 90, 900, 9000, 74667, 608673, …

(\underline{A172199}). The sequence of base 10 squares is not monotonic (for example 1011 < 1020 yet 1011 _{10} 1011 = 1011111 > 1020 _{10} 1020 = 1010200), and contains repetitions. The numbers which are squares in more than one way are

111111111, 111111112, 111111113, 111111114, 111111115, 111111116, 111111117, …,

e.g., 111111111 = 11011 _{10} 11011 = 11111 _{10} 11111 (\underline{A180513}, \underline{A181319}).

We briefly mention two other questions about squares to which we do not know the answer: (i) In base 2, how many square roots does 2^{2k+1} − 1 have? This is a kind of combinatorial covering problem. For k = 0, 1, … the counts are

1, 1, 1, 1, 2, 3, 5, 9, 15, 28, 50, 95, 174, 337, 637, 1231, 2373, 4618, 8974, 17567, 34387, 67561, 132945, 262096, 517373, 1023366, 2025627, 4014861, 7964971, 15814414, 31424805, 62490481, 124330234, 247514283, 492990898, 982307460, 1958093809,

3904594162, 7788271542, 15539347702, 31012331211, …  (19)

(\underline{A191701}). Is there a formula or recurrence for this sequence? (ii) In base b, if we consider all p such that p _{b} p = n, does one of them dominate all the others (in the ≫_{b} sense)? If so the dominating one could be called the “principal” square root.

5 The divisibility poset

One drawback to dismal arithmetic is that there is more than one way to order the dismal numbers, and no ordering is fully satisfactory.

The usual order on the nonnegative integers (< or ≤) is unsatisfactory, since (working in base 10) we have 18 < 25 yet 18 32 = 38 > 25 32 = 35, 32 < 41 yet 32 3 = 32 > 41 3 = 31.

The dominance order (≪_{b}) is more satisfactory, in view of Lemma 6 and the distributive law of Theorem 1, but has the drawback that m divides n does not imply that m ≪_{b} n (e.g., 12|_{10} divides 11|_{10}, yet 11|_{10} ≪_{10} 12|_{10}).

The partial order induced by divisibility (≺_{b}) is worth discussing, as it has some interesting properties and is the best way to look at dismal numbers as long as we are considering only questions of factorization and divisibility. For simplicity we will restrict the discussion to base 10.

Figure 1: Beginning of the divisibility poset (see text for description). The left-hand column gives the rank (\underline{A161813}).

Figure 1 displays the beginning of the Hasse diagram ([21, p. 99]) of this partially ordered set (or poset), and shows all positive numbers with one or two digits, and a few larger numbers. There are too many edges to draw in the diagram, so we will describe them in words.

The multiplicative identity 9, the “zero element” in the poset, is at the base. The other single-digit numbers 8 ≺ 7 ≺ 6 ≺ ⋯ ≺ 1 are above it in the left-hand column, and 8 is joined to 9. The numbers are arranged in rows according to their rank (shown at the extreme left of the diagram). The numbers of rank 1 consist of 8 and the (infinitely many) primes: 19, 29, …, 91, 90, 109, 209, 219, 309, … (\underline{A144171}). All of these are joined to 9. The numbers of rank 2 are 7, 18, 28, …, 81, 80, 108, 119, … (\underline{A144175}), and so on. Every two-digit number of rank h (1 ≤ h ≤ 9) is joined to the single-digit number h − 1 on the left of the diagram., as indicated by the square brackets. A two-digit number to the left of the central column of the pyramid is joined to the number diagonally below it to the left (e.g., 56 is joined to 57), and a two-digit number to the right of the central column is joined to the number diagonally below it to the right (e.g., 65 is joined to 75). A number in the central column is joined to the three numbers immediately below it (e.g., 66 is joined to 67, 77, 76). A number r0 (1 ≤ r ≤ 9) in the right-hand column is joined to the number immediately below it and to the single-digit number r.

Only a few numbers with more than two digits are shown, but a more complete diagram would show for example that 1 is joined to, besides 10 and 11, many other decimal numbers whose digits are 0’s and 1’s, such as 101, 1001, 1011, . . .. all of rank 9. The figure is complete in the sense that all downward joins are shown for all the numbers in the diagram.

One perhaps surprising property of the divisibility poset is that the greatest lower bound (or greatest common divisor) m ∧ n and the least upper bound (or least common multiple) m ∨ n of two dismal numbers m and n need not exist, and so this poset fails to be a lattice [9, Chap. 1]. For example, again working in base 10, the rank 2 numbers 8989 and 9898 are each divisible by (and joined to) the nine primes 909, 919, …, 989. However, these nine primes are incomparable in the ≺_{10} order, so neither 8989 ∧ 9898 nor 909 ∨ 919 exist.

Although greatest common divisors need not exist, we can still define two dismal numbers to be relatively prime if their only common divisor is the unit β.

In the next section we will study the number of divisors function d_{b}(n). Candidates for divisors of n are all numbers m with len_{b}(m) ≤ len_{b}(n). It is therefore appropriate to define the dismal analogue of the Euler totient function, ϕ_{b}(n), to be the number of numbers m with len_{b}(m) ≤ len_{b}(n) which are relatively prime to n. The initial values of ϕ_{2}(n) and ϕ_{10}(n) are shown in Table 4.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ ϕ_{2}(n) & 1 & 2 & 2 & 4 & 6 & 2 & 4 & 8 & 14 & 6 & 14 & 5 & 14 & 5 & 7 & 16 & 30 & 14 & 30 & 12 \\ ϕ_{10}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 9 & 18 & 2 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 90 & 18 \end{matrix}

Table 4: Values of totient functions ϕ_{2}(n) and ϕ_{10}(n) (\underline{A191674}, \underline{A191675}).

6 The number of dismal divisors

Let d_{b}(n) denote the number of dismal divisors of n in base b, and let σ_{b}(n) denote the dismal sum of the dismal divisors of n. These functions are more irregular than their classical analogues, as can be seen from the examples in Table 5, and there are no simple formulas for them. In this section we study some of the properties of d_{b}(n). Note that if r is the largest digit in n, then the smallest divisor of n is r, and the largest divisor is the number obtained by changing all the r’s in n to β’s.

n  divisors (base 10)  d_{10}(n) σ_{10}(n)
1  1, 2, 3, 4, 5, 6, 7, 8, 9  9  9
2  2, 3, 4, 5, 6, 7, 8, 9  8  9
3  3, 4, 5, 6, 7, 8, 9  7  9
4  4, 5, 6, 7, 8, 9  6  9
5  5, 6, 7, 8, 9  5  9
6  6, 7, 8, 9  4  9
7  7, 8, 9  3  9
8  8, 9  2  9
9  9  1  9
10  1, …, 9, 10, 20, …, 90  18  99
11 1, …, 9, rs with 1 ≤ r, s ≤ 9 90  99
12  2, …, 9, 12, 13, …, 19  16  19

Table 5: In base 10, the dismal divisors of the numbers 1 through 12 and the corresponding values of d_{10}(n) and σ_{10}(n) (\underline{A189506}, \underline{A087029}, \underline{A087416}).

A base b dismal prime p has two divisors, b−1 and p, so d_{b}(p) = 2. In the other direction, a divisor of a k-digit number n has at most k digits, so

2 ≤ d_{b}(n) ≤ b^{k} − 1 .  (20)

Base b numbers of the form 111 … 1|_{b} (that is, in which all the base b digits are 1) come close to meeting this upper bound—see Remark (iv) following Theorem 17. We make the following conjectures.

Conjecture 12. In any base b ≥ 3, among all k-digit numbers n, d_{b}(n) has a unique maximum at n = (b^{k} − 1)/(b − 1) = 111 … 1|_{b}.

Conjecture 13. In base 2, among all k-digit numbers n, the maximal value of d_{2}(n) occurs at n = 2^{k} − 2 = 111 … 10|_{2}, and this is the unique maximum for n ≠ 2, 4.

Conjecture 14. In base 2, among all odd k-digit numbers n, d_{2}(n) has a unique maximum at n = 2^{k} − 1 = 111 … 111|_{2}, and if k ≥ 3 and k ≠ 5, the second-largest value of d_{2}(n) occurs at n = 2^{k} − 3 = 111 … 101|_{2}, n = 2^{k} − 2^{k−2} − 1 = 101 … 111|_{2}, and possibly other values of n.

The numerical evidence supporting these conjectures is compelling. For example, in base 10, if we study the sequence d_{10}(n), n ≥ 1 (\underline{A087029}) for n ≤ 10^{6}, and write down d_{10}(n) each time it exceeds d_{10}(m) for all m < n, we obtain the values

9, 18, 90, 180, 819, 1638, 7461, 14922, 67968 ,

(see \underline{A186443}) at these (decimal) values of n:

1, 10, 11, 110, 111, 1110, 1111, 11110, 11111 .

If n is a 5-digit decimal number, the eight largest values of d_{10}(n) are, in decreasing order,

67968, 39624, 21812, 14922, 11202, 9616, 6732, 6570,

at these values of n:

11111, 22222, 33333, 11110, 44444, 12222 or 22220 or 22221, 11011, 10111 or 11101.

The number n = (10^{k} − 1)/9 = 111 … 1|_{10} is a clear winner among all k-digit decimal numbers for k ≤ 5. The data for bases 3 through 9 is equally supportive of Conjecture 12. Likewise, the binary data strongly supports Conjectures 13 and 14—see Table 6 for the initial values of d_{2}(n). Table 6 also shows why k = 5 is mentioned as an exception in Conjecture 14: among 5-digit odd numbers, d_{2}(11011|_{2}) = 4 is the runner-up, ahead of d_{2}(10111|_{2}) = d_{2}(11101|_{2}) = 2.

n (base 2) d_{2}(n) n (base 2) d_{2}(n) n (base 2) d_{2}(n) n (base 2) d_{2}(n)
−  −  1000  4  10000  5  11000  8
1  1  1001  2  10001  2  11001  2
10  2  1010  4  10010  4  11010  4
11  2  1011  2  10011  2  11011  4
100  3  1100  6  10100  6  11100  9
101  2  1101  2  10101  3  11101  2
110  4  1110  6  10110  4  11110  10
111  3  1111  5  10111  2  11111  8

Table 6: In base 2, the number of dismal divisors of the numbers 1 through 31 (\underline{A067399}).

For a more dramatic illustration of Conjectures 12 and 13, see the graphs of sequences \underline{A087029} and \underline{A067399} in [17]. Although the evidence is convincing, we have not, unfortunately, succeeded in proving these conjectures.

We are able to determine the exact values of d_{b}(111 … 111|_{b}) for all b (the conjectural winner for b ≥ 3 and the conjectural winner among odd numbers in the binary case) and d_{2}(111 … 101|_{2}) = d_{2}(101 … 111) (the conjectural runners-up among odd binary numbers of length k > 5).

We begin with a lemma that describes the effect of trailing zeros.

Lemma 15. If the base b expansion of n ends with exactly r ≥ 0 zeros, so that n = mb^{r}, with b ∤ m, then

d_{b}(n) = (r + 1)d_{b}(m) .

Proof. If p _{b} q = m then b ∤ p, b ∤ q, and the r + 1 numbers pb^{i} (0 ≤ i ≤ r) dismally divide n, since

(pb^{i}) _{b} (qb^{r−i}) = n .

Conversely, if p′ _{b} q′ = n then p′ = pb^{i}, q′ = qb^{r−i}, b ∤ p, b ∤ q, for some i with 0 ≤ i ≤ r. So each dismal divisor of m corresponds to exactly r + 1 dismal divisors of n.

For example, in base 10 the dismal divisors of 7 are 7, 8, 9 and the dismal divisors of 700 are 7, 8, 9, 70, 80, 90, 700, 800, 900.

One reason Conjectures 12-14 seem hard to prove is the erratic behavior of d_{b}(n). In contrast to the above lemma, the effect of internal zeros is hard to analyze. Suppose the i-th digit in the b-ary expansion of n is zero. This implies that if n = p _{b} q, then all entries in the i-th column of the long multiplication tableau must be zero, which imposes many constraints on the b-ary expansions of p and q. One would expect, therefore, that changing the zero digit to a larger number—thus weakening the constraints—would always increase the number of divisors of n. Roughly speaking, this is true, but there are many cases where it fails. For example, d_{2}(11111|_{2}) = 8, but d_{2}(11110|_{2}) = 10 (see Table 6, Lemma 15 and Theorem 16). Again, d_{2}(10101|_{2}) = 3, but 10111|_{2} is prime, so d_{2}(10111|_{2}) = 2. In any base b, 11 \underbrace{00 … 0}_{r} |_{b}

has (r + 1)((b − 1)^{2} + (b − 1)) divisors, whereas 11 \underbrace{00 … 0}_{r−1} 1|_{b} has (b − 1)^{3} + (b − 1) divisors,

a smaller number if r is large.

The next result was conjectured by the second author and proved by Richard Schroeppel in 2001 [15]. An alternative proof (via a bijection with a certain class of polyominoes) was given by Frosini and Rinaldi in 2006 [6]. We give a version of Schroeppel’s elegant direct proof, partly because it has never been published, and partly because we will use similar arguments later.

Theorem 16. In base 2, the number of dismal divisors of 111 … 1|_{2} (with k 1’s) is equal to the number of compositions of k into parts of which the first is at least as great as all the other parts.

Proof. Suppose p _{2} q = 111 … 1|_{2} (with k 1’s) where len_{2}(p) = r, len_{2}(q) = k + 1 − r. By examining the long multiplication tableau for p _{2} q, we see that it is also true that p _{2} q′ = 111 … 1|_{2} where q′ = 111 … 1|_{2}, with k +1−r 1’s (for if there is a 1 in each column of the tableau for p _{2} q, that is still true for p _{2} q′). So in order to find all the divisors p of 111 … 1|_{2} (with k 1’s) we may assume that the cofactor q has the form 111 … 1|_{2} (with s 1’s, for some s, 1 ≤ s ≤ k).

We establish the desired result by exhibiting a bijection between the two sets. Let k = c_{1} +c_{2} +⋯+c_{t} be a composition of k in which c_{1} ≥ c_{i} ≥ 1 (2 ≤ i ≤ t). Let ψ(c_{i}) denote the binary vector 000 … 01 with c_{i} − 1 0’s and a single 1. The divisor p corresponding to this composition has binary representation given by the concatenation

1 ψ(c_{2}) ψ(c_{3}) … ψ(c_{r})|_{2} ,  (21)

of length k + 1 − c_{1}. If we set q = 111 … 1|_{2}, of length c_{1}, then p _{2} q = 111 … 1|_{2} (with k 1’s). This follows from the fact that if the binary representation of p contains a string of exactly s 0’s:

… 1 \underbrace{000 … 0}_{s} 1 …,

then, when we form the product p _{2} q, the 1 immediately to the right of these 0’s will propagate leftward to cover the 0’s if and only if len_{2}(q) ≥ s + 1, which is exactly the condition that c_{1} ≥ c_{i} for all i ≥ 2.

For example, the eight compositions of 5 in which no part exceeds the first and the corresponding factorizations p _{2} q of 11111|_{2} are shown in Table 7. Dots have been inserted in p to indicate the division into the pieces ψ(c_{i}).

Composition of 5 divisor p cofactor q

5  1|_{2}  11111|_{2}
41  1.1|_{2}  1111|_{2}
32  1.01|_{2}  111|_{2}
311  1.1.1|_{2}  111|_{2}
221  1.01.1|_{2} 11|_{2}
212  1.1.01|_{2} 11|_{2}
2111  1.1.1.1|_{2} 11|_{2}
11111  1.1.1.1.1|_{2} 1|_{2}

Table 7: Illustrating the bijection used to prove Theorem 16.

Remarks.

(i) Using the bijection defined by (21), the number of 1’s in the binary expansion of the divisor p is equal to the number of parts in the corresponding composition.

(ii) It follows immediately from the interpretation in terms of compositions that the numbers d_{2}(2^{k} − 1) have generating function

∑_{k=1}^{∞} d_{2}(2^{k} − 1) z^{k} = ∑_{l=1}^{∞} \frac{z^{l}}{1 − (z + z^{2} + ⋯ + z^{l})}

= ∑_{l=1}^{∞} \frac{(1 − z)z^{l}}{1 − 2z + z^{l+1}}  (22)

(the index of summation, l, corresponds to the first part in the composition).

(iii) The initial values of this sequence are shown in Table 8. This sequence appears in entries \underline{A007059} and \underline{A079500} in [17], although the indexing is different in each case. The sequence also occurs in at least four other contexts besides the two mentioned in Theorem 16, namely in the enumeration of balanced ordered trees (Kemp [10]), of polyominoes that tile the plane by translation (Beauquier and Nivat [2], Brlek et al. [3]), of Dyck paths (see \underline{A007059}), and in counting solutions to the postage stamp problem (again see \underline{A007059}). The article by Frosini and Rinaldi [6] gives bijections between four of these six enumerations.

\begin{matrix}k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ d_{2}(2^{k} − 1) & 1 & 2 & 3 & 5 & 8 & 14 & 24 & 43 & 77 & 140 & 256 & 472 & 874 & 1628 & 3045 & 5719 \end{matrix}

Table 8: Values of d_{2}(2^{k} − 1).

In this context we should also mention the recent article of Rawlings and Tiefenbruck [19], which, although not directly related to the problems we consider, discusses other connections between the enumeration of compositions, permutations, polyominoes, and binary words.

(iv) The asymptotic behavior of this sequence is quite subtle. From the work of Kemp [10] and Knopfmacher and Robbins [12] it follows that

d_{2}(2^{k} − 1) ∼ \frac{2^{k}}{k log 2} (1 + Θ_{k}), as k → ∞ ,  (23)

where Θ_{k} is a bounded oscillating function with |Θ_{k}| < 10^{−5} (see the proof of Theorem 23 below).

In order to determine d_{b}(111 … 1|_{b}) for bases b > 2, we first classify compositions in which no part exceeds the first according to the number of parts. Let T (k, t) denote the number of compositions of k into exactly t parts (with 1 ≤ t ≤ k) such that no part exceeds the first. Table 9 shows the initial values. This is entry \underline{A184957} in [17]. ^{4}

k\t 1 2 3 4 5 6 7 8
1 1
2 1 1
3 1 1 1
4 1 2 1 1
5 1 2 3 1 1
6 1 3 4 4 1 1
7 1 3 6 7 5 1 1
8 1 4 8 11 11 6 1 1

Table 9: Initial values of of T (k, t), the number of compositions of k into exactly t parts such that no part exceeds the first.

The values of T (k, t) are easily computed via the auxiliary variables γ(k, t, m), which we define to be the number of compositions of k into t parts of which the first part, m, is the greatest (for 1 ≤ t ≤ k, 1 ≤ m ≤ k). We have the recurrence

γ(k, t, m) = ∑_{j=1}^{min{m,k+2−t−m}} γ(k − j, t − 1, m)  (24)

4An array equivalent to this, \underline{A156041}, was contributed to [17] by J. Grahl in 2009 and later studied by A. P. Heinz and R. H. Hardin.

(classifying compositions according to the last part, j), for m > 1, t > 1, t + m < k − 1, with initial conditions

γ(k, t, 1) = δ_{t,k} ,
γ(k, 1, m) = δ_{m,k} ,

where δ_{i,j} = 1 if i = j or 0 if i ≠ j. Then

T (k, t) = ∑_{m=1}^{k+1−t} γ(k, t, m) .  (25)

Since γ(k, t, m) is the coefficient of z^{k} in z^{m}(z + z^{2} + ⋯ + z^{m})^{t−1}, it follows that column t of Table 9 has generating function

∑_{k=1}^{∞} T (k, t)z^{k} = \frac{z^{t−1}}{(1 − z)^{t−1}} ∑_{r=0}^{t−1}(−1)^{r}\binom{t − 1}{r}\frac{z^{r+1}}{1 − z^{r+1}} .  (26)

Since the total number of compositions of k into t parts is \binom{k−1}{t−1}, and in at least a fraction \frac{1}{t} of them the first part is the greatest, we have the bounds

\frac{1}{t} \binom{k − 1}{t − 1} ≤ T (k, t) ≤ \binom{k − 1}{t − 1} .  (27)

Theorem 17.

d_{b}(\frac{b^{k} − 1}{b − 1}) = d_{b}(\underbrace{11 … 1}_{k} |_{b}) = ∑_{t=1}^{k} T (k, t)(b − 1)^{t} .  (28)

Proof. Suppose p _{b} q = \underbrace{11 … 1}_{k} |_{b}. At least one of p and q, say q, must contain only digits 0

and 1 (for if p contains a digit i > 1 and q contains a digit j > 1, then i _{b} j = min{i, j} > 1 will appear somewhere in p _{b} q). As in the proof of Theorem 16 we may assume that this q has the form \underbrace{11 … 1}_{s} |_{b} for some s with 1 ≤ s ≤ k. Suppose p = ∑_{i=0}^{r−1} p_{i}2^{i} with p_{i} ∈ \{0, 1\}

is a divisor of \underbrace{11 … 1}_{k} |_{2}, so that

p _{2} (2^{k+1−r} − 1) = 2^{k} − 1 .  (29)

By Lemma 5, p′ := ∑_{i=0}^{r−1} p_{i}b^{i} is a base b dismal divisor of \frac{b^{k}−1}{b−1} :

p′ _{b} \frac{b^{k+1−r} − 1}{b − 1} = \frac{b^{k} − 1}{b − 1} .  (30)

Furthermore, (30) still holds if any of the p_{i} that are 1 are changed to any digit in the range \{1, 2, …, b − 1\}. Conversely, any base b dismal divisor p′ of \frac{b^{k}−1}{b−1} remains a divisor if all the nonzero digits in the base b expansion of p′ are replaced by 1’s. So each divisor of \underbrace{11 … 1}_{k} |_{2}

with t 1’s corresponds to (b − 1)^{t} divisors of \frac{b^{k}−1}{b−1}. Since there are T (k, t) divisors of 2^{k} − 1 with t 1’s, the result follows.

\begin{matrix}k\b & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 2 & 6 & 12 & 20 & 30 & 42 & 56 & 72 & 90 \\ 3 & 3 & 14 & 39 & 84 & 155 & 258 & 399 & 584 & 819 \\ 4 & 5 & 34 & 129 & 356 & 805 & 1590 & 2849 & 4744 & 7461 \\ 5 & 8 & 82 & 426 & 1508 & 4180 & 9798 & 20342 & 38536 & 67968 \\ 6 & 14 & 206 & 1434 & 6452 & 21830 & 60594 & 145586 & 313544 & 619902 \\ 7 & 24 & 526 & 4890 & 27828 & 114580 & 375954 & 1044246 & 2555080 & 5660208 \end{matrix}

Table 10: Table of d_{b}(11 … 1|_{b}) (with k 1’s), the number of base b dismal divisors of \frac{b^{k}−1}{b−1} (rows: \underline{A002378}, \underline{A027444}, \underline{A186636}; columns \underline{A079500}, \underline{A186523}).

Remarks.

(i) Table 10 shows the initial values of d_{b}(\underbrace{11 … 1}_{k} |_{b}).

(ii) Theorem 17 reduces to Theorem 16 in the case b = 2.

(iii) From (27) and (28) we have

\frac{b^{k} − 1}{k} ≤ d_{b}(\underbrace{11 … 1}_{k} |_{b}) ≤ (b − 1)b^{k−1} .  (31)

For b = 2 we also have the asymptotic estimate (23).

We now study the runners-up in the binary case (among odd numbers of length greater than 5), namely the numbers 111 … 101|_{2} and 101 … 111|_{2}. The simplest way to state the result is to give the generating function.

Theorem 18.

∑_{k=3}^{∞} d_{2}(2^{k} − 3)z^{k} = z + \frac{z^{3}}{1 − z} + ∑_{l=3}^{∞} \frac{(1 − z)^{2}z^{l}}{1 − 2z + z^{l−1} − z^{l} + z^{l+2}}
= z + 2z^{3} + 2z^{4} + 2z^{5} + 4z^{6} + 6z^{7} + 10z^{8} + · · · . (\underline{A188288}) (32)

We will deduce Theorem 18 from Theorem 21 below.

Suppose p _{2} q = 2^{k} − 3, k ≥ 3, where len_{2}(p) = h, len_{2}(q) = l, with h + l = k + 1. In order to find all choices for p, we note that the binary expansions of p and q must end with … 01, and that, as in the proofs of Theorems 16 and 17, we may assume that q = 2^{l} − 3. Our approach is to fix l and allow h to vary. Let M_{h}^{(l)} denote the number of binary numbers p with len_{2}(p) = h such that

p _{2} \underbrace{111 … 101}_{l} |_{2} = \underbrace{111 … 101}_{h+l−1} |_{2} .  (33)

Suppose the binary expansion of p is

1 x_{v} x_{v−1} … x_{3} x_{2} x_{1} 0 1|_{2} ,

where v := h − 3 and the x_{i} are 0 or 1. The long multiplication tableau for (33) implies that the x_{i} must satisfy certain Boolean equations (remember that _{2} is the logical OR; in what follows we will write rather than _{2}). For example, the tableau for l = 4 and h = 9, v = 6 is shown in Figure 2.

1 x_{6}x_{5}x_{4}x_{3}x_{2}x_{1}0 1
2  1 1 0 1
1 x_{6}x_{5}x_{4}x_{3}x_{2}x_{1}0 1
1 x_{6}x_{5}x_{4}x_{3}x_{2}x_{1}0 1
1 x_{6}x_{5}x_{4}x_{3}x_{2}x_{1}0 1
1 1 1 1 1 1 1 1 1 1 0 1

Fig. 2.

By reading down the columns, we obtain the equations

x_{1} x_{3} = 1 ,
x_{1} x_{2} x_{4} = 1 ,
x_{2} x_{3} x_{5} = 1 ,
x_{3} x_{4} x_{6} = 1 ,
x_{5} x_{6} = 1 .

There are M_{9}^{(4)} = 29 solutions (x_{1}, …, x_{6}) to these equations. Table 11 shows the initial values of M_{h}^{(l)}, as found by computer.

\begin{matrix}h\l & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 4 & 1 & 0 & 1 & 1 & 2 & 2 & 2 & 2 \\ 5 & 1 & 0 & 2 & 3 & 3 & 4 & 4 & 4 \\ 6 & 1 & 0 & 2 & 5 & 7 & 7 & 8 & 8 \\ 7 & 1 & 0 & 3 & 9 & 13 & 15 & 15 & 16 \\ 8 & 1 & 0 & 6 & 16 & 24 & 29 & 31 & 31 \\ 9 & 1 & 0 & 10 & 29 & 47 & 56 & 61 & 63 \\ 10 & 1 & 0 & 15 & 53 & 89 & 110 & 120 & 125 \\ 11 & 1 & 0 & 24 & 96 & 170 & 216 & 238 & 248 \\ 12 & 1 & 0 & 40 & 174 & 326 & 422 & 471 & 494 \end{matrix}

Table 11: Table of M_{h}^{(l)} (columns 3 and 4 are \underline{A070550} and \underline{A188223}).

Inspection of the table suggests that the l-th column satisfies the recurrence

M_{h}^{(l)} = M_{h−1}^{(l)} + M_{h−2}^{(l)} + ⋯ + M_{h−l+2}^{(l)} + M_{h−l}^{(l)} + M_{h−l−1}^{(l)} ,  (34)

for l ≥ 3. This will be established in Corollary 22.

We consider the cases h ≤ l+1 and h ≥ l+2 separately. For h ≤ l+1, it is straightforward to show the following:

M_{2}^{(1)} = 0 , M_{h}^{(1)} = 1 (h ≠ 2) , M_{h}^{(2)} = 0 ,  (35)

and, for l ≥ 3, h ≤ l + 1,

M_{h}^{(l)} = \begin{cases}1, & if h = 1; \\ 0, & if h = 2; \\ 2^{h−3}, & if 3 ≤ h ≤ l − 1; \\ 2^{h−3} − 1, & if h = l or l + 1. \end{cases}  (36)

This accounts for the entries in Table 11 that are on or above the line h − l = 1.

We now consider the case 3 ≤ l ≤ h − 2 = v + 1. The multiplication tableau leads to two special equations,

\begin{cases}x_{2} = 1, & if l = 3; \\ x_{1} x_{2} ⋯ x_{l−3} x_{l−1} = 1; & if l ≥ 4, \end{cases}  (37)

and

x_{v−l+3} x_{v−l+4} ⋯ x_{v−1} x_{v} = 1 ,  (38)

together with a family of v − l + 1 further equations, which, if l = 3, are

x_{1} x_{3} = x_{2} x_{4} = ⋯ = x_{v−3} x_{v−1} = x_{v−2} x_{v} = 1 ,  (39)

or, if l ≥ 4, are

x_{1} x_{2} x_{3} ⋯ x_{l−2} x_{l} = 1 ,
x_{2} x_{3} x_{4} ⋯ x_{l−1} x_{l+1} = 1 ,
. . . . . . . . . . . . . . .
x_{v−l+1} x_{v−l+2} x_{v−l+3} ⋯ x_{v−2} x_{v} = 1 .

(40)

The two special equations (37) and (38) involve variables with both low and high indices, which makes induction difficult. We therefore define a simpler system of Boolean equations in which the special constraints apply only to the high-indexed variables.

For l ≥ 3 and n ≥ 1, let D_{n}^{(l)} denote the number of binary vectors x_{1}x_{2} … x_{n} of length n that end with x_{n} = 1, do not contain any substring

\underbrace{00 … 000}_{l} or \underbrace{00 … 010}_{l}

and do not end with

\underbrace{00 … 01}_{l−1} .

Equivalently, D_{n}^{(l)} is the number of solutions to the Boolean equations

x_{1} x_{2} x_{3} ⋯ x_{l−2} x_{l} = 1 ,
x_{2} x_{3} x_{4} ⋯ x_{l−1} x_{l+1} = 1 ,
. . . . . . . . . . . . . . .
x_{n−l} x_{n−l+1} x_{n−l+2} ⋯ x_{n−3} x_{n−1} = 1 ,  (41)

and

x_{n−l+2} x_{n−l+3} ⋯ x_{n−2} x_{n−1} = 1 , x_{n} = 1 .  (42)

We also set D_{0}^{(l)} := 1.

Theorem 19. For l ≥ 3, n ≥ 1, there is a one-to-one correspondence between binary vectors of length n satisfying the D_{n}^{(l)} equations and compositions of n into parts taken from the set

{1, 2, …, l − 2, l, l + 1} .  (43)

Proof. We exhibit a bijection between the two sets. Let c be a composition n = c_{1}+c_{2}+⋯+c_{r} into parts from (43). For 1 ≤ i ≤ l − 2, let ψ(i) := 00 … 01, of length i and ending with a single 1, let ψ(l) := 00 … 011, of length l, let ψ(l + 1) := 00 … 0011, of length l + 1, and let

ψ(c) := ψ(c_{1}) ψ(c_{2}) … ψ(c_{r}) ,  (44)

a binary vector of length n. Note that the ψ(i) for 1 ≤ i ≤ l − 2 contain runs of at most l − 3 zeros. Runs of l − 2 or l − 1 zeros in ψ(c) are therefore followed by two ones. So conditions (41) and (42) are satisfied. Conversely, given a binary vector satisfying the D_{n}^{(l)} equations, we can decompose it into substrings ψ(i) by reading it from left to right.

\begin{matrix}n\l & 3 & 4 & 5 & 6 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 2 & 2 & 2 \\ 3 & 2 & 3 & 4 & 4 \\ 4 & 4 & 6 & 7 & 8 \\ 5 & 6 & 11 & 14 & 15 \\ 6 & 9 & 20 & 27 & 30 \\ 7 & 15 & 36 & 51 & 59 \\ 8 & 25 & 65 & 98 & 115 \end{matrix}

Table 12: Table of D_{n}^{(l)} (the columns are \underline{A006498}, \underline{A079976}, \underline{A079968}, \underline{A189101}).

The generating function for the D_{n}^{(l)} follows immediately from the theorem:

Corollary 20. For l ≥ 3, the numbers D_{n}^{(l)} have generating function

D^{(l)}(z) := ∑_{n=0}^{∞} D_{n}^{(l)}z^{n} = \frac{1}{1 − (z + z^{2} + ⋯ + z^{l−2} + z^{l} + z^{l+1})}
= \frac{1 − z}{1 − 2z + z^{l−1} − z^{l} + z^{l+2}} .  (45)

Table 12 shows the initial values of D_{n}^{(l)}, computed using the generating function. The l = 4 and l = 5 columns are in [17] as entries \underline{A079976} and \underline{A079968}, taken from a paper by D. H. Lehmer on enumerating permutations (π_{1}, …, π_{n}) with restrictions on the displacements π_{i} − i ([14]; see also [11]).

We now express the numbers M_{h}^{(l)} in terms of the D_{n}^{(l)}. We consider the l values x_{1}, …, x_{l} in a solution to the M_{h}^{(l)} equations, and classify them according to the number of leading zeros. There are just l −1 possibilities, as shown in Table 13, and in each case the M_{h}^{(l)} equations reduce to an instance of the D_{n}^{(l)} equations. For example, if x_{1} = 1 the M_{h}^{(l)} equations reduce to an instance of the D_{h−3}^{(l)} equations. (E.g., if l = 5 and we set x_{1} = 1, equations (37), (38), (40) become x_{v−2} x_{v−1} x_{v} = 1, x_{2} x_{3} x_{4} x_{6} = 1, …, x_{v−4} x_{v−3} x_{v−2} x_{v} = 1, which, if we subtract 1 from each subscript, are the equations for D_{v}^{(5)}, that is, D_{h−3}^{(5)}.)

Setting, in M_{h}^{(l)},  leads to
x_{1} = 1  D_{h−3}^{(l)}
x_{1} = 0, x_{2} = 1  D_{h−4}^{(l)}
x_{1} = x_{2} = 0, x_{3} = 1  D_{h−5}^{(l)}
… …
x_{1} = ⋯ = x_{l−4} = 0, x_{l−3} = 1  D_{h−l+1}^{(l)}
x_{1} = ⋯ = x_{l−3} = 0, x_{l−2} = x_{l−1} = 1 D_{h−l−1}^{(l)}
x_{1} = ⋯ = x_{l−2} = 0, x_{l−1} = x_{l} = 1 D_{h−l−2}^{(l)}

Table 13: Expressing M_{h}^{(l)} in terms of D_{n}^{(l)}.

We have therefore shown that for l ≥ 3, h ≥ 3,

M_{h}^{(l)} = D_{h−3}^{(l)} + D_{h−4}^{(l)} + D_{h−5}^{(l)} + ⋯ + D_{h−l+1}^{(l)} + D_{h−l−1}^{(l)} + D_{h−l−2}^{(l)} .  (46)

Table 14 illustrates (46) in the case l = 4.

From (46) and Corollary 20, and taking into account the values of M_{h}^{(l)} for h < 3, we obtain:

\begin{matrix}h & M_{h}^{(4)} & D_{h−3}^{(4)} & D_{h−5}^{(4)} & D_{h−6}^{(4)} \\ 3 & 1 & 1 & − & − \\ 4 & 1 & 1 & − & − \\ 5 & 3 & 2 & 1 & − \\ 6 & 5 & 3 & 1 & 1 \\ 7 & 9 & 6 & 2 & 1 \\ 8 & 16 & 11 & 3 & 2 \\ 9 & 29 & 20 & 6 & 3 \end{matrix}

Table 14: Illustrating M_{h}^{(4)} = D_{h−3}^{(4)} + D_{h−5}^{(4)} + D_{h−6}^{(4)}.

Theorem 21. For l ≥ 3,

M^{(l)}(z) := ∑_{h=1}^{∞} M_{h}^{(l)}z^{h} = z + (z^{3} + z^{4} + ⋯ + z^{l−1} + z^{l+1} + z^{l+2}) D^{(l)}(z)

= \frac{z(1 − z)}{1 − (z + z^{2} + ⋯ + z^{l−2} + z^{l} + z^{l+1})}

= \frac{z(1 − z)^{2}}{1 − 2z + z^{l−1} − z^{l} + z^{l+2}} .  (47)

It is now straightforward to obtain the recurrence for M_{h}^{(l)} from the generating function in the second line of the display. We omit the proof.

Corollary 22. For l ≥ 3, M_{h}^{(l)} satisfies the recurrence (34) with initial conditions (35), (36).

We can now give the proof of Theorem 18. From the definition of M_{h}^{(l)}, we have

d_{2}(2^{k} − 3) = ∑_{l=1}^{k} M_{k−l+1}^{(l)} .

That is, d_{2}(2^{k} − 3) is the sum of the coefficient of z^{k} in M^{(1)}(z), the coefficient of z^{k−1} in M^{(2)}(z), …, and the coefficient of z^{1} in M^{(k)}(z). In other words, d_{2}(2^{k} − 3) is the coefficient of z^{k} in

M^{(1)}(z) + zM^{(2)}(z) + z^{2}M^{(3)}(z) + ⋯ + z^{k−1}M^{(k)}(z) ,

and now (32) follows from M^{(1)}(z) = z + z^{3}/(1 − z), M^{(2)}(z) = 0, and Theorem 21. This completes the proof of Theorem 18.

Remark. The l = 3 column of the M_{h}^{(l)} table (Table 11) is an interesting sequence in its own right.^{5} To analyze it directly, first consider the system of simultaneous Boolean equations

x_{1} x_{2} = x_{2} x_{3} ⋯ x_{n−1} x_{n} = 1 ,  (48)

5It is entry \underline{A070550} in [17], which contains a comment by Ed Pegg, Jr., that it arises in the analysis of Penney’s game.

for n ≥ 2, involving a chain of linked pairs of variables. An easy induction shows that the number of solutions is the Fibonacci number F_{n+2} (cf. \underline{A000045}).^{6} Second, the equations for M_{h}^{(3)}, (38) and (39), break up into two disjoint chains like (48), and we find that

M_{h}^{(3)} = \begin{cases}F_{(n−2)/2}F_{n/2}, & if n is even; \\ F_{(n−3)/2}F_{(n+1)/2}, & if n is odd. \end{cases}  (49)

From (49) we can derive the recurrence M_{h}^{(3)} = M_{h−1}^{(3)} + M_{h−3}^{(3)} + M_{h−4}^{(3)} and the generating function

M^{(3)}(z) = \frac{z(1 − z)}{1 − z − z^{3} − z^{4}} ,  (50)

in agreement with (34) and (47).

The final result in this section will describe the asymptotic behavior of the sequence d_{2}(2^{k} − 3). When investigating Conjectures 13 and 14, we observed that among all numbers n with k binary digits, the number 2^{k} − 1 was the clear winner, with values close to the estimate (23). The runners-up, a long way behind, were 2^{k} − 3 and 2^{k} − 2^{k−2} − 1 (and sometimes other values of n), all with the same number of dismal divisors, for which the number of dismal divisors appeared to be converging to one-fifth of the number of divisors of the winner, or in other words it appeared that

\frac{d_{2}(2^{k} − 3)}{d_{2}(2^{k} − 1)} → \frac{1}{5}, as k → ∞ .  (51)

We will now establish this from the generating function (32).

Theorem 23.

d_{2}(2^{k} − 3) ∼ \frac{2^{k}}{5k log 2} (1 + Θ̄_{k}), as k → ∞ ,  (52)

where Θ̄_{k} is a bounded oscillating function with |Θ̄_{k}| < 10^{−5}.

Proof. Our proof is modeled on Knopfmacher and Robbins’s proof [12] of (23), which uses the method of Mellin transforms as presented by Flajolet, Gourdon, and Dumas [5]. We will indicate how the Knopfmacher-Robbins proof can be reworded so as to establish (23) and (52) simultaneously.

Knopfmacher and Robbins work, not with (22), but with

f(z) := ∑_{l=1}^{∞} \frac{(1 − z)z^{l}}{1 − 2z + z^{l}} ,  (53)

which is the generating function for the number of compositions of n into parts of which the first is strictly greater than all the other parts (\underline{A007059}). Equations (22) and (53) basically differ just by a factor of z. Then [12] shows that the coefficient of z^{n} in f(z) is

\frac{2^{n−1}}{n log 2}(1 + Θ) ,  (54)

6This result could also be obtained by the Goulden-Jackson cluster method, as implemented by Noonan and Zeilberger [8], [16].

for some small oscillating function Θ, which implies (23).

So as to have a function with the same form as (53), we consider, not (32), but

f(z) := ∑_{l=2}^{∞} \frac{(1 − z)^{2}z^{l}}{1 − 2z + z^{l} − z^{l+1} + z^{l+3}} ,  (55)

and we will show that the coefficient of z^{n} is

\frac{2^{n+1}}{5n log 2}(1 + Θ) ,  (56)

for some (different) small oscillating function Θ, which implies (52). We can change the lower index of summation in (55) from 2 to 1, since the l = 1 term is the generating function for the Padovan sequence (\underline{A000931}), which grows at a much slower rate than (56).

In what follows, we simply record how the expressions in Knopfmacher and Robbins’s proof [12] need to be modified so as to apply simultaneously to (53), which we refer to as case I, and (55), which we call case II. We follow Knopfmacher and Robbins’s notation, except that we use j and m as local variables, rather than k, to avoid confusion with the k in the statement of the theorem. Some typographical errors in [12] have been silently corrected.

Let ρ_{j} denote the smallest root of the denominator of the j-th summand in f(z) that lies between 0 and 1. Then

ρ_{j} = \frac{1}{2} (1 + τ2^{−j} + O(j2^{−2j})) ,  (57)

where τ = 1 (case I) or 5/8 (case II).

Let q_{n,j} denote the coefficient of z^{n} in the j-th summand in f(z). Then

q_{n,j} ≈ 2^{n−j−ϵ} (1 − \frac{τ}{2^{j}} )^{n}
≈ 2^{n−j−ϵ} e^{−τn/2^{j}} ,  (58)

where ϵ = 1 in case I or 2 in case II. Next, f_{n}, the coefficient of z^{n} in f(z), is

2^{n−ϵ} ( ∑_{j=2}^{∞} 2^{−j}e^{−τn/2^{j}} + o(1)) .

Let

g(x) := ∑_{j=2}^{∞} 2^{−j}e^{−τx/2^{j}} .

The Mellin transform of g(x) is

g^{∗}(s) := \frac{1}{τ^{s}} \frac{2^{2(s−1)}}{1 − 2^{s−1}} Γ(s), 0 < ℜ(s) < 1 .

To compute f_{n} we use (following Knopfmacher and Robbins) the Mellin inversion formula

g(x) = \frac{1}{2πi} ∫_{1/2−i∞}^{1/2+i∞} x^{−s}g^{∗}(s) ds .

Now g^{∗}(s)x^{−s} has a simple pole at s = 1 + χ_{m}, for each m ∈ ℤ, where χ_{m} = 2πim/ log 2, with residue

− \frac{1}{x log 2} \frac{Γ(1 + χ_{m}) e^{−2πim log_{2} x}}{τ^{1+χ_{m}}} .

After combining the contributions from all the poles, we have

f_{n} = 2^{n−ϵ} ∑_{m=−∞}^{∞} \frac{1}{n log 2} \frac{Γ(1 + 2πim/ log 2) e^{−2πim log_{2} n}}{τ^{1+2πim/ log 2}} .  (59)

The term for m = 0 dominates, and we obtain the desired results (54) and (56).

7 The sum of dismal divisors

We briefly discuss the dismal sum-of-divisors function σ_{b}(n) (see \underline{A188548}, \underline{A190632}, and \underline{A087416} for bases 2, 3, and 10).

Theorem 24. In any base b ≥ 2, if len_{b}(n) = k, then

n ≤ σ_{b}(n) ≤ b^{k} − 1 ,  (60)

and σ_{b}(n) = n if and only if n ≡ b − 1 (mod b).

Proof. The first assertion follows because n divides itself, and no divisor has length greater than k. If n ≢ b − 1 (mod b) then since b − 1 is the multiplicative unit, σ_{b}(n) ≠ n. Suppose n ≡ b − 1 (mod b) and p is a divisor of n, with say p _{b} q = n. Both p and q must end with β. From the long multiplication tableau, p ≪_{b} n, so p _{b} n = n, and therefore σ_{b}(n) = n.

In ordinary arithmetic, a number n is perfect if its sum of divisors is 2n. In dismal arithmetic, n _{b} n = n. So the second part of the theorem might, by a stretch, be interpreted as saying that the numbers congruent to b−1 (mod b) are the base b perfect dismal numbers.

In base 2, then, σ_{2}(n) = n if and only if n is odd. Examination of the data shows that, if n is even, with len_{2}(n) = k, often σ_{2}(n) takes its maximal value, 2^{k} − 1. Table 15 shows the first few exceptions, which are characterized in the next theorem.

Theorem 25. Suppose n = 2^{r}m with r ≥ 1, m odd, and len_{2}(n) = k. Then

σ_{2}(n) = 2^{k} − 1 = \underbrace{11 … 1}_{k} |_{2}

unless the binary expansion of m contains a run of more than r consecutive zeros.

Proof. Since m is odd, σ_{2}(m) = m. Therefore

σ_{2}(n) = m|_{2} _{2} m0|_{2} _{2} m00|_{2} _{2} ⋯ _{2} m \underbrace{00 … 0}_{r} |_{2} ,

and any string \underbrace{00 … 0}_{i} 1 in m will become \underbrace{11 … 1}_{i} 1 in σ_{2}(n) unless i exceeds r.

\begin{matrix}n & σ_{2}(n) & n & σ_{2}(n) \\ 10010 & 11011 & 1001010 & 1101111 \\ 100010 & 110011 & 1001110 & 1101111 \\ 100110 & 110111 & 1010010 & 1111011 \\ 110010 & 111011 & 1100010 & 1110011 \\ 1000010 & 1100011 & 1100110 & 1110111 \\ 1000100 & 1110111 & 1110010 & 1111011 \\ 1000110 & 1100111 & . . . . . . & . . . . . . \end{matrix}

Table 15: Even numbers n such that σ_{2}(n) is not of the form 11 … 1|_{2} (\underline{A190149}–\underline{A190151}). Both n and σ_{2}(n) are written in base 2.

The first entry in Table 15 is explained by the fact that n = 10010|_{2}, r = 1, m = 1001|_{2}, and m contains a run of two zeros.

We also considered two other possible definitions of perfect numbers: (i) n is perfect in base b ≥ 3 if the dismal sum of the dismal divisors of n is equal to 2 _{b} n. We leave it to the reader to verify that for this to happen, b must be 3, and then n is perfect if and only if n ≡ 2 mod 4. (ii) n is perfect in base b ≥ 2 if the dismal sum of the dismal divisors of n different from n is equal to n. But here n cannot be b − 1, so b − 1 is a divisor, and n ends with b − 1. This implies that n has no divisors of length len_{b}(n) except n itself, so the sum cannot equal n, and therefore no such n exists.

We end this section with a conjecture (see \underline{A186442}):

Conjecture 26. For all n > 1, d_{10}(n) < σ_{10}(n).

8 Dismal partitions

Since n _{b} n = n, it only makes sense to consider partitions into distinct parts (otherwise every number has infinitely many different partitions). We define p_{b}(n) to be the number of ways of writing

n = m_{1} _{b} m_{2} _{b} ⋯ _{b} m_{l} ,  (61)

for some l ≥ 1 and distinct positive integers m_{i}, without regard to the order of summation. We set p_{b}(0) = 1 by convention.

For example, p_{3}(7) = p_{3}(21|_{3}) = 22, since (working in base 3) 21 is equal to 21 _{3} any subset of {20, 11, 10, 1} (16 solutions), 20 _{3} 11 _{3} any subset of {10, 1} (4 solutions), and 20 _{3} 1 _{3} any subset of {10} (2 solutions), for a total of 22 solutions. Remarks.

(i) Permuting the digits of n does not change p_{b}(n).

(ii) Any zero digits in n can be ignored. If n′ is the base b number obtained by dropping any n_{i}’s that are zero, p_{b}(n′) = p_{b}(n).

(iii) Although we will not make any use of it, there is a generating function for the p_{b}(n) analogous to that for the classical case. If we interpret z^{m} _{b} z^{n} to mean z^{m _{b} n}, then we have the formal power series

1 + p_{b}(1)z + p_{b}(2)z^{2} + p_{b}(3)z^{3} + ⋯ = (1 + z) _{b} (1 + z^{2}) _{b} (1 + z^{3}) _{b} · · · .

(iv) The sequences p_{2}(n) and p_{10}(n) form entries \underline{A054244} and \underline{A087079} in [17], contributed by the second author in 2000 and 2003, respectively.

In the remainder of this section we index the digits of n by {1, 2, …, n}, in order to simplify the discussion of subsets of these indices.

Theorem 27. If n = n_{1}n_{2} … n_{k}|_{2}, n_{i} ∈ {0, 1}, and the binary weight of n is w, then p_{2}(n) is equal to the number of set-covers of a labeled w-set by nonempty sets (cf. \underline{A003465}, [4, p. 165]), that is,

p_{2}(n) = \frac{1}{2} ∑_{i=0}^{w}(−1)^{w−i}\binom{w}{i}2^{2^{i}} .  (62)

Proof. From Remark (ii), p_{2}(n) = p_{2}(\underbrace{11 … 1}_{w} |_{2}). There is an obvious one-to-one corre-

spondence between collections of distinct nonempty subsets of {1, …, w} whose union is \{1, …, w\} and sets of distinct nonzero binary vectors whose dismal sum is \underbrace{11 … 1}_{w} |_{2}.

Theorem 28. If n = n_{1}n_{2} … n_{k}|_{b}, 0 ≤ n_{i} ≤ b − 1, then

p_{b}(n) = \frac{1}{2} ∑_{S⊆\{1,...,k\}}(−1)^{|S|} 2^{∏_{i}(n_{i}+ϵ_{i})} ,  (63)

where ϵ_{i} = 0 if i ∈ S, ϵ_{i} = 1 if i ∉ S.

Proof. The set Ω_{n} of x ≪_{b} n is a partially ordered set (with respect to the operator ≪_{b}) with Möbius function given by [21, §3.8.4]

μ(x_{1} … x_{k}|_{b}, y_{1} … y_{k}|_{b}) = \begin{cases}(−1)^{∑_{i}(y_{i}−x_{i})}, & if y_{i} − x_{i} = 0 or 1 for all i; \\ 0, & otherwise. \end{cases}  (64)

Every subset of Ω_{n} \ {0} has dismal sum equal to some number ≪_{b} n, so we have

∑_{x≪_{b}n} p_{b}(x) = 2^{∏_{i}(n_{i}+1)−1} .

From the Möbius inversion formula [21, §3.7.1] we get

p_{b}(n) = \frac{1}{2} ∑_{S⊆\{1,...,k\}}(−1)^{|S|} 2^{∏_{i∈S} n_{i} ∏_{j∉S}(n_{j}+1)} ,

which implies (63).

Theorem 28 reduces to Theorem 27 if all n_{i} are 0 or 1.

Corollary 29. Suppose n = n_{1}n_{2} … n_{k}|_{b}, and let y be the ordinary product n_{1}×n_{2}×⋯×n_{k}. Then p_{b}(n) is divisible (in ordinary arithmetic!) by 2^{y−1}.

Corollary 30. For a single-digit number n = n_{1}|_{b}, p_{b}(n) = 2^{n_{1}−1}. For a two-digit number n = n_{1}n_{2}|_{b},

p_{b}(n) = 2^{(n_{1}+1)(n_{2}+1)−2} − 2^{n_{1}n_{2}−1}(2^{n_{1}} − 1)(2^{n_{2}} − 1) .  (65)

Eq. (65) was found by Wasserman [17, entry A087079].

It follows from the above discussion that in any base b, the only numbers n such that p_{b}(n) = 1 are 0|_{b}, 1|_{b}, 10|_{b}, 100|_{b}, 1000|_{b}, …, and that all other numbers n have the property that the dismal sum of the numbers x ≪_{b} n is n. These two classes might be called “additive primes” and “additive perfect numbers.”

9 Conclusion and future explorations

We have attempted to show that dismal arithmetic, despite its simple definition, is worth studying for the interesting problems that arise. We have left many questions unanswered: the “prime number theorem” of Conjecture 10, the questions about the numbers of divisors stated in Conjectures 12-14 (in particular, is it true that 11 … 1|_{10} has more divisors than any other base 10 number with the same number of digits?), the base 10 dismal analog of d(n) ≤ σ(n) (Conjecture 26), and the two questions about dismal squares at the end of §4—in particular, is there a recurrence for the sequence (19)? There are numerous other questions that we have not investigated (for example, if x ≪_{b} y, what can be said about x _{b} y?).

We have made no mention of the complexity of deciding if a number is a dismal prime, or of finding dismal factorizations. In base 2 such questions reduce to solving a set of simultaneous quadratic Boolean equations, where a typical equation might be

(x_{0} _{2} y_{4}) _{2} (x_{1} _{2} y_{3}) _{2} ⋯ _{2} (x_{4} _{2} y_{0}) = 0 (or 1) .

This becomes a question about the satisfiability of a complicated Boolean expression, and is likely to be hard to solve in general [7].

While we have focused on the cases b = 2 and b = 10, it would be nice to better understand the qualitative differences across a wider range of bases. For example, while b = 2 is a kind of Boolean arithmetic, does b = 3 correspond to a three-valued logic? Do odd b behave differently from even b? More generally, what other interesting mathematical structures might be modeled by dismal arithmetic?

10 Acknowledgments

We thank Adam Jobson for computing an extended version of Table 11, which was helpful in guessing the recurrence (34). Most of the our computations were programmed in C++, Fortran, Lisp, and Maple. We also made use of Mathematica’s SatisfiabilityCount command, and we thank Michael Somos for telling us about it. We are also grateful to Doron Zeilberger for telling us about his work with John Noonan [16] on implementing the Goulden-Jackson cluster method. The OEIS [17] repaid us several times during the course of this work by providing valuable hints, notably from the entries \underline{A007059}, \underline{A067399}, \underline{A070550}, \underline{A079500}, \underline{A156041}, and \underline{A164387}.

References

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[7] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, 1979.

[8] I. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, NY, 1983.

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[14] D. H. Lehmer, Permutations with strongly restricted displacements, in P. Erdős, A. Rényi, and V. T. Sós, eds., Combinatorial Theory and its Applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, 1970, pp. 755–770.

[15] M. LeBrun and R. Schroeppel, Postings to the Math-Fun Mailing List, December, 2001. See entry \underline{A048888} in [17].

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    web site: http://www.math.rutgers.edu/∼zeilberg/gj.html.

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    http://oeis.org, 2011.

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[19] D. Rawlings and M. Tiefenbruck, Consecutive patterns: from permutations to columnconvex polyominoes and back, Electron. J. Combin 17 (2010), #R62.

[20] J. Richter-Gebert, B. Sturmfels, and T. Theobald, First steps in tropical geometry, in G. L. Litvinov and V. P. Maslov, eds., Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics 377, American Mathematical Society, 2005, pp. 289–317.

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2010 Mathematics Subject Classification: Primary 06A06, 11A25, 11A63; Secondary 11N37. Keywords: Carryless arithmetic, squares, primes, divisors, compositions, partitions, asymptotic expansions, Mellin transform.

(Concerned with sequences \underline{A000045}, \underline{A000695}, \underline{A000931}, \underline{A002378}, \underline{A003465}, \underline{A006498}, \underline{A007059}, \underline{A009994}, \underline{A009996}, \underline{A027444}, \underline{A048888}, \underline{A054244}, \underline{A067139}, \underline{A067398}, \underline{A067399}, \underline{A070550}, \underline{A079500}, \underline{A079968}, \underline{A079976}, \underline{A087019}, \underline{A087029}, \underline{A087052}, \underline{A087079}, \underline{A087097}, \underline{A087416}, \underline{A087636}, \underline{A087984}, \underline{A088469}, \underline{A088470}, \underline{A088471}, \underline{A088472}, \underline{A088473}, \underline{A088475}, \underline{A088476}, \underline{A088477}, \underline{A088478}, \underline{A088479}, \underline{A088480}, \underline{A088481}, \underline{A144171}, \underline{A144175}, \underline{A156041}, \underline{A161813}, \underline{A162672}, \underline{A164387}, \underline{A169912}, \underline{A171000}, \underline{A171004}, \underline{A171120}, \underline{A171786}, \underline{A171787}, \underline{A171788}, \underline{A171789}, \underline{A171794}, \underline{A171795}, \underline{A171796}, \underline{A171818}, \underline{A172199}, \underline{A180513}, \underline{A181319}, \underline{A184957}, \underline{A186413}, \underline{A186442}, \underline{A186443}, \underline{A186523}, \underline{A186636}, \underline{A188223}, \underline{A188288}, \underline{A188548}, \underline{A189101}, \underline{A189506}, \underline{A189788}, \underline{A190149}, \underline{A190150}, \underline{A190151}, \underline{A190632}, \underline{A190820}, \underline{A191420}, \underline{A191674}, \underline{A191675}, \underline{A191701}, \underline{A191752}, and \underline{A191753}.)

Received July 7 2011; revised version received October 25 2011. Published in Journal of Integer Sequences, November 21 2011.

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