Computing the Inverses, their Power Sums, and Extrema for Euler’s Totient and

Other Multiplicative Functions

Max A. Alekseyev Department of Mathematics The George Washington University 2115 G St. NW Washington, DC 20052 USA

                     maxal@gwu.edu

Abstract

We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power sums (e.g., cardinality) or extrema, without direct enumeration of the inverses. We illustrate our algorithm with Euler’s totient function φ and the k-th power sum of divisors σ_{k} . For example, we can establish that the number of solutions to σ_{1} x 10^{1000} is 15, 512, 215, 160, 488, 452, 125, 793, 724, 066, 873, 737, 608, 071, 476, while it is intractable to iterate over the actual solutions.

1 Introduction

A value of a multiplicative function f on a positive integer n equals the product of its values on the prime powers in the prime factorization of n. That is, if n p_{1}^{e_{1}} p_{2}^{e_{2}} p_{m}^{e_{m}}, where p_{1} p_{2} p_{m} are primes and e_{1}, e_{2}, …, e_{m} are positive integers, then

f n _{i 1}^{m} f p_{i}^{e_{i}} .

In particular, f 1 1. Famous examples of multiplicative functions include τ n , the number of divisors of n (with τ p^{e} e 1); σ_{k} n , the k-th power sum of divisors of n (with σ_{k} p^{e} \frac{p^{k e 1} 1}{p^{k} 1} ); and Euler’s totient function φ n (with φ p^{e} p 1 p^{e 1}).

In the present work, we propose a generic algorithm for computing the set of inverses (full pre-image) f^{1} n of a multiplicative function f for a given integer n under the following assumptions: (i) there are finitely many prime powers p^{e} with f p^{e} n, and we can compute them all; (ii) the prime factorization of n is known, otherwise it may be a bottleneck to obtain (e.g., Contini et al. [2] proved that computing φ^{1} can be used for factoring semiprime integers with little overhead). In particular, our algorithm is well applicable for computing φ^{1} n and σ_{k}^{1} n with k 0,^{1} which we will use for illustration purposes. While computing inverses of Euler’s totient function φ was studied to some extent [3, 2, 1], computing inverses of other multiplicative functions such as σ_{k} , to the best of our knowledge, was not addressed in the literature. Our algorithm may be viewed as a generalization and streamlining of the “intelligent exhaustive search” for φ^{1} n in [2].

We present an underlying idea of the algorithm in the elegant form of formal Dirichlet series, which allow us to easily extend it to computing certain functions of the inverses, such as their power sums (including cardinality as the 0-th power sum) or extrema, without and faster than direct enumeration of the inverses. For example, our algorithm can establish that

σ_{1}^{1} 10^{1000} 15, 512, 215, 160, 488, 452, 125, 793, 724, 066, 873, 737, 608, 071, 476. (1)

2 Formal Dirichlet series framework

From now on, we assume that f is a fixed multiplicative function.

We find it convenient to define binary multiplication and addition operations on sets of positive integers as follows: U V u v u U, v V and U V U V. Equipped with these operations the set _{fin} ℤ_{0} of finite^{2} subsets of positive integers forms a commutative semiring (with the additive identity and the multiplicative identity 1 ) and allows us to consider formal Dirichlet series with coefficients from this semiring.^{3}

Theorem 1. We have the following identity for formal Dirichlet series of variable s over the semiring _{fin} ℤ_{0} , , :

_{n 1} \frac{f^{1} n}{n^{s}} _{prime p} _{e 0}^{}\frac{p^{e}}{f p^{e} ^{s}}.  (2)

1While τ n can be viewed as a special case of σ_{k} n with k 0, the assumption (i) does not hold in this case already for n 2.

2The combinatorial identities (2), (3), (6), (7) proved in this section hold in the case of infinite pre-images as well, while our restriction to the finite case is dictated purely by computational needs.

3To emphasize that the defined and are semiring operations, we use their big versions (in place of more traditional Σ and ) to denote the corresponding series summation and product operators. Some background information on formal Dirichlet series over semirings is given in the Appendix.

For a fixed positive integer n and every divisor d n, we further have

f^{1} d Coeff_{d^{s}} _{prime p} _{e f p^{e} n} \frac{p^{e}}{f p^{e} ^{s}}.  (3)

Proof. Multiplicativity of f implies that if n f m and m p_{1}^{e_{1}} p_{2}^{e_{2}} p_{k}^{e_{k}}, where p_{1} p_{2} p_{k} are primes, then n f p_{1}^{e_{1}} f p_{2}^{e_{2}} f p_{k}^{e_{k}} . It follows that n is the product of factors of the form f p^{e} , where p is a prime and e is a positive integer, and no two such factors share the same p. In other words,

f^{1} n _{f p_{1}^{e_{1}} f p_{k}^{e_{k}} n} _{i 1}^{k} p_{i}^{e_{i}} ,  (4)

where the sum is taken over various tuples of primes p_{1} p_{2} p_{k} (with arbitrary k 0) and various positive integer exponents e_{1}, e_{2}, …, e_{k} that satisfy f p_{1}^{e_{1}} f p_{2}^{e_{2}} f p_{k}^{e_{k}} n. Multiplying (4) by n^{s}, we get

\frac{f^{1} n}{n^{s}} _{f p_{1}^{e_{1}} f p_{k}^{e_{k}} n} _{i 1}^{k} \frac{p_{i}^{e_{i}}}{f p_{i}^{e_{i}} ^{s}}.

Summing over n 1, we obtain

_{n 1} \frac{f^{1} n}{n^{s}} _{{p_{1} p_{2} p_{k} \atop e_{1},e_{2},...,e_{k} 0}} _{i 1}^{k} \frac{p_{i}^{e_{i}}}{f p_{i}^{e_{i}} ^{s}} _{prime p} _{e 0}^{}\frac{p^{e}}{f p^{e} ^{s}},

which proves (2).

We remark that terms with e 0 in (2) represent multiplicative identities (i.e., \frac{p^{0}}{f p^{0} ^{s}} \frac{1}{1^{s}} for any prime p), while for a prime power p^{e} with e 0, f p^{e} may participate in a factorization of n only if f p^{e} n. Hence, to obtain the full pre-image f^{1} n from (2) for a given n, we can restrict our attention only to such prime powers:

f^{1} n Coeff_{n^{s}} _{prime p} _{e f p^{e} n} \frac{p^{e}}{f p^{e} ^{s}}.  (5)

We further remark that for every divisor d n, the coefficients of d^{s} in the series in the right hand side of (5) and (2) coincide, which implies formula (3).

Let X, , be a commutative semiring. A mapping C  _{fin} ℤ_{0} , ,  X, ,

is a weak homomorphism if for any U, V _{fin} ℤ_{0} , we have C U V C U C V whenever the sets U and V are element-wise coprime (i.e., gcd u, v 1 for any u U and v V), and C U V C U C V whenever U, V are disjoint. It is easy to see that if C is a homomorphism (i.e., C U V C U C V and C U V C U C V hold unconditionally), then it is also a weak homomorphism.

Theorem 2. Let X, , be a commutative semiring and C _{fin} ℤ_{0} , , X, , be a weak homomorphism, then

_{n 1} \frac{C f^{1} n}{n^{s}} _{prime p} _{e 0}^{}\frac{C p^{e}}{f p^{e} ^{s}}.  (6)

Furthermore, for a fixed positive integer n and every divisor d n,

C f^{1} d Coeff_{d^{s}} _{prime p} _{e f p^{e} n} \frac{C p^{e}}{f p^{e} ^{s}}.  (7)

Proof. We remark that the sets inside the product in (4) are coprime, while the products inside the sum are disjoint. Since C is a weak homomorphism, we have

C f^{1} n _{f p_{1}^{e_{1}} f p_{k}^{e_{k}} n} _{i 1}^{k} C p_{i}^{e_{i}} ,

which further implies identity (6). Formula (7) is derived from (6) with the same arguments we used to derive (3) from (2).

Formula (7) under appropriate choice of the weak homomorphism C and its codomain X, , allows us to efficiently compute certain functions of the inverses without their direct enumeration. In the next section we give some particular examples.

3 Examples of weak homomorphisms

Our first, rather trivial example is given by X, , _{fin} ℤ_{0} , , with C being the identity homomorphism. In this case, formulae (6) and (7) simply represent the original formulae (2) and (3) for the full pre-images. We will keep this trivial example in mind to fit computation of full pre-images into our generic algorithm.

Our second example is given by X, , ℤ_{0}, max, , which is a commutative semiring of nonnegative integers with a binary maximum operation (i.e., u v max u, v ) and the standard integer multiplication. The mapping C U max U 0 , giving the maximum element of U or 0 for U , represents a homomorphism between _{fin} ℤ_{0} , , and ℤ_{0}, max, .

Similarly, the mapping C U  min U  represents a homomorphism between
fin ℤ_{0} , , and the commutative semiring X, ,  ℤ_{0}  , min, , where m

m and m m m for any element m X (i.e., represents an additive identity).

An example of a weak homomorphism, which is not a homomorphism, is given by X, , ℤ_{0}, , , a semiring of nonnegative integers with the standard integer addition and multiplication, and C_{q} U _{u U} u^{q}, where q is a fixed nonnegative integer. In particular, C_{0} U U represents the cardinality of a set U, while C_{1} U is the sum of elements of U.

4 Algorithm for computing C f^{1} n

In addition to a multiplicative function f, we now fix a weak homomorphism C from fin ℤ_{0} , , to a commutative semiring X, , . To compute C f^{1} n for a given integer n with a known prime factorization, we iteratively compute the right hand side of (7) restricted to the terms with denominators d^{s} for d n. This computation naturally splits into three major steps outlined below.

Step 1. From the prime factorization of n, we easily compute the set of its divisors D. Clearly, D τ n .

Step 2. We compute the atomic series

L_{p} _{e f p^{e} n}\frac{C p^{e}}{f p^{e} ^{s}} _{d D} \frac{A_{d}}{d^{s}}

for every prime p that admits at least one^{4} integer e 0 with f p^{e} n. Here

A_{d} _{e f p^{e} d}C p^{e} .  (8)

We remark that finiteness of full pre-images of the function f implies that the number of the atomic series is finite, since for each atomic series L_{p}, some positive power of p must belong to the finite set _{d D} f^{1} d . Internally it is convenient to store each such atomic series L_{p} as an associative array d A_{d} indexed by elements d D.

Step 3. We multiply the constructed atomic series L_{p_{1}}, L_{p_{2}}, …, L_{p_{ℓ}} and compute partial products P_{0} \frac{C 1}{1^{s}}, P_{1} P_{0} _{D} L_{p_{1}}, …, P_{ℓ} P_{ℓ 1} _{D} L_{p_{ℓ}}, where _{D} denotes the result of restricted to the terms with denominators d^{s} for d D. Each multiplication is computed with the formula:

_{d D} \frac{A_{d}}{d^{s}} _{D} _{d D} \frac{B_{d}}{d^{s}}  _{d D} \frac{_{t d}A_{t} B_{d t}}{d^{s}} .  (9)

That is, if the associative arrays d A_{d} and d B_{d} represent L_{p_{j 1}} and the partial product P_{j}, then we compute the partial product P_{j 1} as an associative array d _{t d}A_{t} B_{d t}.

For every d D, the coefficient of d^{s} in the final product P_{ℓ} gives us C f^{1} d . In particular, C f^{1} n Coeff_{n^{s}} P_{ℓ}.

While Step 1 of the algorithm is rather trivial and takes O τ n arithmetic operations on integers of length O log n , Step 2 is specific to a particular function f and illustrated with some examples in the next section. Below we analyze the time complexity of the generic Step 3.

4We remark that if there is no such e 0, then L_{p} \frac{C 1}{1^{s}} represents the identity for -multiplication of formal Dirichlet series.

Theorem 3. Let n be an integer and D be the set of divisors of n. Given ℓ atomic series for C f^{1} n , their _{D}-product can be computed with O ℓ τ n ^{2} operations in X, , .

Proof. Computation of the _{D}-product consists of ℓ iterative computations of pairwise D-multiplications defined by (9). Since each operand of such _{D}-multiplication contains O τ n terms, its computation takes O τ n ^{2} operations and .

As we will see in the next section, for Euler’s totient function φ we have ℓ τ n . In particular, Step 3 in computation of the size or extrema of φ^{1} n takes O τ n ^{3} arithmetic operations on integers of length O log n , which all can be done in O τ n ^{3} log^{2} n time. Combining with results of the next section, we obtain that Steps 1-3 in this case can be done in O τ n log^{2} n τ n ^{2} log^{4} n . Similarly, for the function σ_{k} , we have ℓ τ n log_{2} n , thus computation of the size or extrema of σ_{k}^{1} n can be done in O τ n log^{3} n τ n ^{2} log^{4} n time.

5 Computation of atomic series

As we explained above, our algorithm is generic and works for any multiplicative function f, provided that we can construct atomic series L_{p_{1}}, …, L_{p_{ℓ}}. To construct them, we need to determine suitable primes p and compute the corresponding series coefficients A_{d} defined by (8). Below we describe such computation in details for the functions φ and σ_{k} . We remark that in both cases we rely on primality testing, which for a number with n bits can be done in time O log^{6 ϵ} n for any ϵ 0 [4]. Using probabilistic primality test (e.g., Miller–Rabin test [5]) can save a factor of log^{3} n.

5.1 Euler’s totient function

Our goal is to find the prime powers p^{e} such that φ p^{e} n. Since for e 0, φ p^{e} p 1 p^{e 1}, the divisibility φ p^{e} n implies that p 1 divides n and e ν_{p} n 1, where ν_{p} n is the p-adic valuation of n (i.e., the maximum integer t such that p^{t} but not p^{t 1} divides n). So we need to compute the set S p p 1 D and p is prime , which can be done by going over the elements d of D and testing if p d 1 is prime. The set S gives us the indices of the atomic series. For every prime p S, we compute the corresponding atomic series:

L_{p} \frac{C 1}{1^{s}} _{e 1}^{ν_{p} n 1} \frac{C p^{e}}{p 1 p^{e 1} ^{s}} .

Example. For n 12 with the set of divisors D 1, 2, 3, 4, 6, 12 , we obtain the set of primes S 2, 3, 5, 7, 13 . If C U is a weak homomorphism into X, , ℤ_{0}, , giving the sum of the elements of U, then the corresponding atomic series are

L_{2}  \frac{1}{1^{s}}  \frac{2}{1^{s}}  \frac{4}{2^{s}}  \frac{8}{4^{s}}
\frac{3}{1^{s}}  \frac{4}{2^{s}}  \frac{8}{4^{s}} ,
L_{3}  \frac{1}{1^{s}}  \frac{3}{2^{s}}  \frac{9}{6^{s}} ,
L_{5}  \frac{1}{1^{s}}  \frac{5}{4^{s}} ,
L_{7}  \frac{1}{1^{s}}  \frac{7}{6^{s}} ,
L_{13}  \frac{1}{1^{s}}  \frac{13}{12^{s}} .

The partial _{D}-products in this case are P_{0} \frac{C 1}{1^{s}} \frac{1}{1^{s}} and

\begin{matrix}P_{1} &  & P_{0} _{D} L_{2} &  & \frac{3}{1^{s}}  \frac{4}{2^{s}}  \frac{8}{4^{s}} , \\ P_{2} &  & P_{1} _{D} L_{3} &  & \frac{3}{1^{s}}  \frac{13}{2^{s}}  \frac{20}{4^{s}}  \frac{27}{6^{s}}  \frac{36}{12^{s}} , \\ P_{3} &  & P_{2} _{D} L_{5} &  & \frac{3}{1^{s}}  \frac{13}{2^{s}}  \frac{35}{4^{s}}  \frac{27}{6^{s}}  \frac{36}{12^{s}} , \\ P_{4} &  & P_{3} _{D} L_{7} &  & \frac{3}{1^{s}}  \frac{13}{2^{s}}  \frac{35}{4^{s}}  \frac{48}{6^{s}}  \frac{127}{12^{s}}, \\ P_{5} &  & P_{4} _{D} L_{13} &  & \frac{3}{1^{s}}  \frac{13}{2^{s}}  \frac{35}{4^{s}}  \frac{48}{6^{s}}  \frac{166}{12^{s}}. \end{matrix}

The coefficient of 12^{s} in P_{5} gives the sum of φ^{1} 12 13, 21, 26, 28, 36, 42 .

To analyze the running time of this algorithm, let T_{C} m be the maximum time required to compute C t for a positive integer t having at most m bits.

Theorem 4. Given an integer n and the set of its divisors D, the atomic series for C φ^{1} n can be computed in time O τ n log n log^{5 ϵ} n T_{C} 2 log n for any ϵ 0.

Proof. The proposed algorithm performs O τ n primality tests of positive integers below n 1, each of which takes time O log^{6 ϵ} n . For every identified prime p, it further computes ν_{p} n 2 O log n values of C on singleton sets with elements below p^{ν_{p} n 1} n n 1 , which takes time O log n T_{C} 2 log n .

We remark that for C computing the size or extrema of φ^{1} n , we have T_{C} m O 1 so that the time complexity for computing the atomic series becomes simply O τ n log^{6 ϵ} n .

5.2 Power sum of divisors

Let k be a positive integer. Our goal is to find the prime powers p^{e} such that σ_{k} p^{e} n, i.e., σ_{k} p^{e} d for some d D. Since σ_{k} p^{e} 1 p^{k} p^{2k} p^{ek}, we have p^{ek} d 1 p ^{ek} or p^{ek} d 1 1 p ^{ek}, implying that p d 1 ^{1 ek} . We let d run over D and e run incrementally from 1 to \frac{log_{2} d 1}{k} . For each such pair d, e , we test whether p d 1 ^{1 ek} is prime and whether \frac{p^{e 1 k} 1}{p^{k} 1} d. If both these conditions hold, we have σ_{k} p^{e} d and add the term \frac{C p^{e}}{d^{s}} to L_{p}. Here we assume that initially all L_{p} \frac{C 1}{1^{s}}, and only those L_{p} that were enriched with additional terms in the above process represent the atomic series.

Example. Let us compute the minimum of σ_{1}^{1} n for n 42. So we use the mapping C U min U into the semiring X, , ℤ_{0} , min, . We compute the set D 1, 2, 3, 6, 7, 14, 21, 42 of divisors of n and use it as described above to determine that only prime powers p^{e} 2, 2^{2}, 5, 13, 41 satisfy σ_{1} p^{e} n. They give rise to the following atomic series:

\begin{matrix}L_{2} &  & \frac{1}{1^{s}}  \frac{2}{3^{s}}  \frac{4}{7^{s}} , \\ L_{5} &  & \frac{1}{1^{s}}  \frac{5}{6^{s}} , \\ L_{13} &  & \frac{1}{1^{s}}  \frac{13}{14^{s}} , \\ L_{41} &  & \frac{1}{1^{s}}  \frac{41}{42^{s}} . \end{matrix}

The partial _{D}-products in this case are P_{0} \frac{C 1}{1^{s}} \frac{1}{1^{s}} and

\begin{matrix}P_{1} &  & P_{0} _{D} L_{2} &  & \frac{1}{1^{s}}  \frac{2}{3^{s}}  \frac{4}{7^{s}} , \\ P_{2} &  & P_{1} _{D} L_{5} &  & \frac{1}{1^{s}}  \frac{2}{3^{s}}  \frac{5}{6^{s}}  \frac{4}{7^{s}}  \frac{20}{42^{s}} , \\ P_{3} &  & P_{2} _{D} L_{13} &  & \frac{1}{1^{s}}  \frac{2}{3^{s}}  \frac{5}{6^{s}}  \frac{4}{7^{s}}  \frac{13}{14^{s}}  \frac{20}{42^{s}} , \\ P_{4} &  & P_{3} _{D} L_{41} &  & P_{3}. \end{matrix}

The coefficient of 42^{s} in P_{4} P_{3} gives the minimum of σ_{1}^{1} 42 20, 26, 41 , which is 20.

As above, let T_{C} m be the maximum time required to compute C t for a positive integer t having at most m bits. Theorem 5. Given an integer n and the set of its divisors D, the atomic series for C σ_{k}^{1} n can be computed in time O τ n log n log^{6 ϵ} n T_{C} log n for any ϵ 0. Proof. The proposed algorithm performs O τ n log n arithmetic operations and primality tests on positive integers below n, each of which takes time O log^{6 ϵ} n . For every identified prime power p^{e}, it further computes C p^{e} , which takes time O T_{C} log n .

As above, for C computing the size or extrema of σ_{k}^{1} n , we have T_{C} m O 1 so that the time complexity for computing the atomic series becomes simply O τ n log^{7 ϵ} n .

6 Examples in the OEIS

The Online Encyclopedia of Integer Sequences [6] contains a number of sequences, for which the proposed algorithm can compute many terms:^{5}

φ^{1} n! σ_{1}^{1} n! φ^{1} 10^{n} σ_{1}^{1} 10^{n} σ_{1}^{1} p_{n}#
size \underline{A055506} \underline{A055486} \underline{A072074} \underline{A110078} \underline{A153078}
min \underline{A055487} \underline{A055488} \underline{A072075} \underline{A110077} \underline{A153076}
max \underline{A165774} \underline{A055489} \underline{A072076} \underline{A110076} \underline{A153077}

In particular, the value of σ_{1}^{1} 10^{1000} in (1) represents the 1000-th term of the sequence \underline{A110078}.

5Some of the terms in these sequences were computed by Ray Chandler.

7 Acknowledgments

The work was supported by the National Science Foundation under Grant No. IIS-1253614.

Appendix. Formal Dirichlet series

As we are not aware if Dirichlet series were considered over commutative semirings such as fin ℤ_{0} , , before, we feel obliged to briefly overview their properties.

Let X, , be a commutative semiring. A formal^{6} Dirichlet series over is an infinite sequence x_{1}, x_{2}, … of elements of X, which we find convenient to represent as

\frac{x_{1}}{1^{s}} \frac{x_{2}}{2^{s}} _{i 1}^{}\frac{x_{i}}{i^{s}} ,

where s is a formal variable.

It is easy to check that formal Dirichlet series over form a commutative semiring under the addition and multiplication binary operations inherited from as follows:

_{i 1}^{}\frac{x_{i}}{i^{s}}  _{i 1}^{}\frac{y_{i}}{i^{s}}  _{i 1}^{}\frac{x_{i} y_{i}}{i^{s}}

and

_{i 1}^{}\frac{x_{i}}{i^{s}}  _{i 1}^{}\frac{y_{i}}{i^{s}}  _{i 1}^{}\frac{_{j i}x_{j} y_{i j}}{i^{s}} .

These operations can be also viewed as an extension of the operations defined on single terms:

\frac{x}{m^{s}} \frac{y}{m^{s}} \frac{x y}{m^{s}}

and

\frac{x}{m^{s}} \frac{y}{n^{s}} \frac{x y}{m n ^{s}}.

If ϵ and ι represent respectively the additive and multiplicative identities in , then

_{i 1}^{}\frac{ϵ}{i^{s}}

and

\frac{ι}{1^{s}} _{i 2}^{}\frac{ϵ}{i^{s}}

represent respectively the additive and multiplicative identities in the semiring of formal Dirichlet series over . It is often convenient to omit terms with coefficients equal ϵ, e.g., we can simply write \frac{ι}{1^{s}}.

We denote the coefficient of \frac{1}{d^{s}} in a formal Dirichlet series F by Coeff_{d^{s}} F.

6The term “formal” in the description of Dirichlet series and its variable s reflects the fact that they do not take any values (thus a formal Dirichlet series is not a function of s). In contrast, conventional Dirichlet series (e.g., Riemann zeta function ζ s _{i 1}^{}\frac{1}{i^{s}}) are often viewed as functions of a real or complex variable s.

References

[1] R. Coleman, On the image of Euler’s totient function, preprint, 2009. Available at

   http://arxiv.org/abs/0910.2223.

[2] S. Contini, E. Croot, and I. Shparlinski, Complexity of inverting the Euler function, Math. Comp. 75 (2006), 983–996.

[3] H. Gupta, Euler’s totient function and its inverse, Indian J. Pure Appl. Math. 12 (1981), 22–30.

[4] H. W. Lenstra, Jr. and C. Pomerance, Primality testing with Gaussian periods, preprint,

   2005. Available at https://math.dartmouth.edu/~carlp/PDF/complexity12.pdf.

[5] M. O. Rabin, Probabilistic algorithm for testing primality, J. Number Theory 12 (1980), 128–138.

[6] The OEIS Foundation, The On-Line Encyclopedia of Integer Sequences, (2016), published electronically at http://oeis.org.

2010 Mathematics Subject Classification: Primary 11Y16; Secondary 11A05, 11Y55, 11Y70, 30B50. Keywords: multiplicative function, Euler’s totient function, sum of divisors, inverse function, Dirichlet series.

(Concerned with sequences \underline{A055486}, \underline{A055487}, \underline{A055488}, \underline{A055489}, \underline{A055506}, \underline{A072074}, \underline{A072075}, \underline{A072076}, \underline{A110076}, \underline{A110077}, \underline{A110078}, \underline{A153076}, \underline{A153077}, \underline{A153078}, and \underline{A165774}.)

Received November 23 2015; revised version received April 27 2016. Published in Journal of Integer Sequences, May 11 2016.

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