A Family of Sequences Generating Smith Numbers

Amin Witno Department of Basic Sciences Philadelphia University 19392 Jordan

                       awitno@gmail.com

Abstract

Infinitely many Smith numbers can be constructed using sequences which involve repunits. We provide an alternate method of construction by introducing a generalization of repunits, resulting in new Smith numbers whose decimal digits are composed of zeros and nines.

1 Introduction

We work with natural numbers in their decimal representation.

Let S(N) denote the “digital sum” (the sum of the base-10 digits) of the number N and let S_{p}(N) denote the sum of all the digital sums of the prime factors of N, counting multiplicity. For instance, S(2013) = 2 + 0 + 1 + 3 = 6 and, since 2013 = 3 × 11 × 61, we have S_{p}(2013) = 3 + 1 + 1 + 6 + 1 = 12. The quantity S_{p}(N) does not seem to have a name in the literature, so let us just call S_{p}(N) the p-digit sum of N.

The natural number N is called a Smith number when N is composite and S(N) = S_{p}(N). For example, since 22 = 2 × 11, we have both S(22) = 4 and S_{p}(22) = 4. Hence, 22 is a Smith number—thus named by Wilansky [4] in 1982.

Several different ways of constructing Smith numbers are already known. (See our 2010 article [5], for instance, which contains a brief historical account on the subject as well as a list of references.

The first method of construction that actually produces infinitely many Smith numbers was given in 1987 by McDaniel [2], and later another by Costello and Lewis [1]. The key idea that makes these methods successful is based on a pair of simple identities which first appeared in an article by Oltikar and Wayland [3], applying for all natural numbers N:

1. S(10N) = S(N),

2. S_{p}(10N) = S_{p}(N) + 7.

With these two identities in mind, suppose that we have found a number N such that S(N) − S_{p}(N) is a nonnegative multiple of 7, say S(N) − S_{p}(N) = 7c. Then, we will have S(N ×10^{c}) = S(N) and S_{p}(N ×10^{c}) = S_{p}(N)+7c. In other words, S(N ×10^{c}) = S_{p}(N ×10^{c}) and hence, N × 10^{c} is a Smith number.

Therefore, if we can find a sequence which contains infinitely many numbers N such that S(N) − S_{p}(N) is a nonnegative multiple of 7, then we will have infinitely many Smith numbers of the form N × 10^{c}, for a suitable value of c that varies with N.

Since the quantity S(N) − S_{p}(N) will occur quite frequently in our discussion, let us give this expression a convenient notation.

Definition 1. With the domain of all numbers N ≥ 2, we define the function Δ(N) by

Δ(N) = S(N) − S_{p}(N).

Using this definition, we want to find infinitely many N where Δ(N) is nonnegative and a multiple of 7. It seems proper for us to reintroduce McDaniel’s construction of such numbers N first, before we proceed with our own version.

McDaniel’s sequence is given in the form 9R_{n}t_{n}, where R_{n} = (10^{n} − 1)/9—also known as the n-th repunit—and where t_{n} is to be selected from the set M = {2, 3, 4, 5, 7, 8, 15}. These seven elements of M have been cleverly chosen so that they have distinct p-digit sums modulo 7.

A previously known fact concerning repunits [2, Theorem 2] is that S(9R_{n}t) = 9n for all n ≥ 1 and for any natural number t < 10^{n}. In particular, S(9R_{n}t) = 9n for any t ∈ M with n ≥ 2 and, by inspection, with n = 1 as well.

Based on this fact, the number t_{n} in the sequence 9R_{n}t_{n} is defined to be the unique element of M for which S_{p}(t_{n}) ≡ 9n − S_{p}(9R_{n}) (mod 7), for then

Δ(9R_{n}t_{n}) = S(9R_{n}t_{n}) − S_{p}(9R_{n}t_{n})
= 9n − S_{p}(9R_{n}) − S_{p}(t_{n})
≡ 0 (mod 7).

And in order to see that Δ(9R_{n}t_{n}) ≥ 0 for every n ≥ 1, we state the next theorem, also by McDaniel [2, Theorem 1], right after the following definition.

Definition 2. For any number N ≥ 2, the function Ω(N) equals the number of prime factors of N, counting multiplicity.

Theorem 3. Let D(N) denote the number of digits in N. Then S_{p}(N) < 9D(N)−0.54Ω(N).

Applying the theorem, plus the fact that S_{p}(t) ≤ 8 for all t ∈ M and the fact that Ω(9R_{n}) ≥ 3, we get

S_{p}(9R_{n}t_{n}) = S_{p}(9R_{n}) + S_{p}(t_{n}) < 9n − 0.54 × 3 + 8 ≤ 9n + 6.

This last inequality implies that Δ(9R_{n}t_{n}) ≥ −6 and therefore, since Δ(9R_{n}t_{n}) is constructed to be a multiple of 7, we have Δ(9R_{n}t_{n}) ≥ 0 as desired. Finally then, for each n ≥ 1, the Smith number that can be created is 9R_{n}t_{n}×10^{c}, with the value of c for which 7c = Δ(9R_{n}t_{n}).

In a similar way, Costello and Lewis constructed Smith numbers using another sequence in the form 9R_{n}11^{x_{n}} for some computed value of x_{n} between 0 and 6 that will make Δ(9R_{n}11^{x_{n}}) a nonnegative multiple of 7. Although techniquely new, the sequence 9R_{n}11^{x_{n}} is essentially the same as that of McDaniel, with only the set M now replaced by {11^{x} | 0 ≤ x ≤ 6}, but which plays the same role as does M, i.e., to contain 7 elements with distinct p-digit sums modulo 7.

In this article, we shall demonstrate how to construct Smith numbers using new sequences which still resemble that of McDaniel. However, this time we will modify the repunits R_{n} by allowing zero digits in them in a certain way, and then adjust the set M of the seven multipliers accordingly.

2 New Constructions

Definition 4. For every pair (k, n) of natural numbers, we define the number P_{k,n} by

P_{k,n} = ∑_{i=0}^{n−1} 10^{ki}.

For example, P_{3,5} = 1, 001, 001, 001, 001. Note that, in general, we have S(P_{k,n}) = n for any pair (k, n) and that, in particular, P_{1,n} = R_{n}.

For a fixed k ≥ 2, we shall build a sequence in the form 9P_{k,n}t_{k,n}, where t_{k,n} is to be determined in such a way that Δ(9P_{k,n}t_{k,n}) will be a nonnegative multiple of 7 for infinitely many values of n. We will achieve this goal through a series of results, involving the numbers P_{k,n}, in what follows.

Theorem 5. Let k and n be two given natural numbers. If n is a multiple of d, then we have the identity P_{k,d} × P_{kd,n/d} = P_{k,n}. Thus, if d divides n, then P_{k,d} divides P_{k,n}.

Proof. We set x = 10^{k} in the following series of identities.

P_{k,d} × P_{kd,n/d} = ( ∑_{i=0}^{d−1} x^{i}) ( ∑_{j=0}^{n/d−1} x^{dj})

= (1 + x + x^{2} + ⋯ + x^{d−1}) (1 + x^{d} + x^{2d} + ⋯ + x^{n−d})

= ∑_{h=0}^{n−1} x^{h} = P_{k,n}.

Theorem 6. For any values of k ≥ 1 and n ≥ 2, we have Ω(P_{k,n}) ≥ Ω(n).

Proof. Let p_{1}, p_{2}, …, p_{r} be prime numbers, not assumed distinct, such that n = ∏_{i=1}^{r} p_{i}. Let us set n_{j} = ∏_{i=j}^{r} p_{i}, where 1 ≤ j ≤ r. In particular, n_{1} = n. By Theorem 5, we state that P_{k,n_{j}} is a factor of P_{k,n_{j−1}} for all j in the range 2 ≤ j ≤ r. It follows that P_{k,n_{1}} must have at least r prime factors, i.e., Ω(P_{k,n}) ≥ r = Ω(n).

Theorem 7. Fix a natural number k ≥ 2. Suppose that t is the sum of k powers of 10, written t = ∑_{j=1}^{k} 10^{e_{j}}, such that the set {e_{j} | 1 ≤ j ≤ k} serves as a complete residue system modulo k. Then S(9P_{k,n}t) = 9kn for every n ≥ 1.

Proof. Observe that for n ≥ 1,

P_{k,n} × t = (∑_{i=0}^{n−1} 10^{ki}) (∑_{j=1}^{k} 10^{e_{j}})

= ∑_{j=1}^{k} ∑_{i=0}^{n−1} 10^{ki+e_{j}}.

So we see that P_{k,n}t is the sum of kn powers of 10, necessarily all of which are distinct since the exponents e_{j} are distinct modulo k. This means that the digits in the number P_{k,n}t are composed of only zeros and ones—with exactly kn ones. We then have S(9P_{k,n}t) = 9kn as claimed.

Remark 8. Incidentally, Theorem 7 remains valid with k = 1, giving the result S(9R_{n}t) = 9n, where t is just any power of 10. Secondly, in general, the number t given in the theorem is always divisible by R_{k}. To see why, we use the fact that 10^{k} ≡ 1 (mod R_{k}) to obtain

t ≡ ∑_{j=1}^{k} 10^{e_{j} mod k} = R_{k} ≡ 0 (mod R_{k}).

Theorem 9. For a fixed k ≥ 2, suppose that M_{k} is a set of 7 natural numbers t with the following two conditions.

1. Every element t ∈ M_{k} can be expressed as t = ∑_{j=1}^{k} 10^{e_{j}}, where {e_{j} | 1 ≤ j ≤ k} is a complete residue system modulo k.

2. The set {S_{p}(t) | t ∈ M_{k}} is a complete residue system modulo 7.

Then, for each n ≥ 1, there exists a unique t_{k,n} ∈ M_{k} such that Δ(9P_{k,n}t_{k,n}) is a multiple of 7. Furthermore, there exist infinitely many values of n ≥ 1 for which Δ(9P_{k,n}t) ≥ 0 for all t ∈ M_{k}.

Proof. Let t ∈ M_{k}. We have S(9P_{k,n}t) = 9kn according to Theorem 7, so

Δ(9P_{k,n}t) = 9kn − S_{p}(9P_{k,n}) − S_{p}(t).

Hence, for a fixed n ≥ 1, there is exactly one element t_{k,n} ∈ M_{k} for which Δ(9P_{k,n}t_{k,n}) is a multiple of 7, i.e., the element t_{k,n} with S_{p}(t_{k,n}) ≡ 9kn − S_{p}(9P_{k,n}) (mod 7).

Next, it is not hard to see that D(9P_{k,n}) = D(P_{k,n}) = k(n − 1) + 1, recalling that the function D(N) counts the number of digits in N. We then use Theorem 3 to obtain

S_{p}(9P_{k,n}t) = S_{p}(9P_{k,n}) + S_{p}(t)
< 9(kn − k + 1) − 0.54Ω(P_{k,n}) + S_{p}(t)
= S(9P_{k,n}t) − 9k + 9 − 0.54Ω(P_{k,n}) + S_{p}(t).

With the assumption that k ≥ 2, we have that

Δ(9P_{k,n}t) > 9 + 0.54Ω(P_{k,n}) − S_{p}(t).

Therefore, we will have Δ(9P_{k,n}t) ≥ 0 for all t ∈ M_{k} given that, say, Ω(P_{k,n}) ≥ 2S_{p}(t′), for some t′ ∈ M_{k} with S_{p}(t′) ≥ S_{p}(t) for all t ∈ M_{k}. The claim is now clear since Theorem 6 implies that Ω(P_{k,n}) can be made arbitrarily large for infinitely many values of n, in particular when n has sufficiently many prime divisors.

Thus according to Theorem 9, we have infinitely many n where Δ(9P_{k,n}t_{k,n}) = 7c with c ≥ 0, each of which is associated with the Smith number 9P_{k,n}t_{k,n} × 10^{c}—provided that a set M_{k} such as described in the theorem has been found. For k = 2, to begin with, we propose the following set as a valid M_{2} example.

M_{2} = {11, 1001, 100001, 10^{15} + 1, 10^{21} + 1, 10^{23} + 1, 10^{35} + 1}.

To help see that the hypothesis of Theorem 9 is satisfied, we provide in Table 1 the prime factorization for each t ∈ M_{2}, in order to evaluate S_{p}(t).

Table 1: The elements t ∈ M_{2} and their p-digit sums.

t  Factorization of t into primes  S_{p}(t) S_{p}(t) mod 7
11  11  2  2
1001  7 × 11 × 13  13  6
100001 11 × 9091  21  0
10^{15} + 1 7 × 11 × 13 × 211 × 241×  53  4

2161 × 9091

10^{21} + 1 7 × 7 × 11 × 13 × 127 × 2689× 117  5

459691 × 909091

10^{23} + 1 11 × 47 × 139 × 2531× 122 3

549797184491917

10^{35} + 1 11 × 9091 × 909091 × 4147571× 155  1

265212793249617641

Moreover, to illustrate the construction of the Smith numbers, let us consider P_{2,7}, which factors into three primes:

P_{2,7} = 1010101010101 = 239 × 4649 × 909091.

Here we have S(9P_{2,7}t_{2,7}) = 9 × 2 × 7 = 126, S_{p}(P_{2,7}) = 65, and

Δ(9P_{2,7}t_{2,7}) = 126 − (6 + 65 + S_{p}(t_{2,7})) = 55 − S_{p}(t_{2,7}).

We are led to t_{2,7} = 1001, where S_{p}(1001) = 13 ≡ 55 (mod 7). In the end we have Δ(9P_{2,7} × 1001) = 55 − 13 = 7 × 6, generating the Smith number

9 × 1010101010101 × 1001 × 10^{6} = 9, 099, 999, 999, 999, 909, 000, 000.

Further computational results show that M_{k} can be produced for at least seven more values of k:

M_{3} ={111, 10101, 100011, 110001, 10100001, 100000011, 110000000001},
M_{4} ={1111, 101101, 1001011, 1101001, 10000111, 11100001, 10000000001101},
M_{5} ={11111, 1011101, 10011011, 11011001, 100010111, 111010001, 100000011101},
M_{6} ={111111, 10111101, 100111011, 110111001, 1000110111, 1110110001, 100100011011},
M_{7} ={1111111, 101111101, 1001111011, 1101111001, 10001110111, 11101110001,
1001001011011},
M_{8} ={11111111, 1011111101, 10011111011, 11011111001, 100011110111, 111011110001,
10010011011011},
M_{9} ={111111111, 10111111101, 100111111011, 110111111001, 1000111110111,
1110111110001, 100100111011011}.

We leave it to the reader to verify that each set M_{k} given above, where 3 ≤ k ≤ 9, indeed meets the hypothesis of Theorem 9 and therefore can be used to generate Smith numbers in conjunction with the sequence P_{k,n}.

An apparent future research problem, as we conclude, is to investigate whether or not a suitable M_{k} exists for every k > 9, or perhaps for infinitely many values of k, without resorting to brute force computation.

3 Acknowledgments

The author is truly grateful to the anonymous referee for proofreading this article, starting from its earliest version, and for giving an extensive list of corrections and suggestions, each one of which is sincerely appreciated.

References

[1] P. Costello and K. Lewis, Lots of Smiths, Math. Mag. 75 (2002), 223–226.

[2] W. L. McDaniel, The existence of infinitely many k-Smith numbers, Fibonacci Quart. 25 (1987), 76–80.

[3] S. Oltikar and K. Wayland, Construction of Smith numbers, Math. Mag. 56 (1983), 36–37.

[4] A. Wilansky, Smith numbers, Two-Year College Math. J. 13 (1982), 21.

[5] A. Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22

   (2010), 97–101. Available at http://projecteuclid.org/euclid.mjms/1312233139.

2010 Mathematics Subject Classification: Primary 11A63. Keywords: Smith number, repunit.

(Concerned with sequences \underline{A006753} and \underline{A002275}.)

Received September 29 2012; revised versions received December 21 2012; March 21 2013. Published in Journal of Integer Sequences, March 25 2013.

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