A034695 Tau_6 (the 6th Piltz divisor function), the number of ordered 6-factorizations of n; Dirichlet convolution of number-of-divisors function (A000005) with A007426.

1, 6, 6, 21, 6, 36, 6, 56, 21, 36, 6, 126, 6, 36, 36, 126, 6, 126, 6, 126, 36, 36, 6, 336, 21, 36, 56, 126, 6, 216, 6, 252, 36, 36, 36, 441, 6, 36, 36, 336, 6, 216, 6, 126, 126, 36, 6, 756, 21, 126, 36, 126, 6, 336, 36, 336, 36, 36, 6, 756, 6, 36, 126, 462, 36, 216, 6, 126

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 29 and 38
• Leveque, William J., Fundamentals of Number Theory. New York:Dover Publications, 1996, ISBN 9780486689067, p .167-Exercise 5.b.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
• E. Pérez Herrero, Piltz Divisor functions (1), Psychedelic Geometry Blogspot, Dec 21 2009.
• E. Pérez Herrero, Piltz Divisor functions (2), Psychedelic Geometry Blogspot, Dec 24 2009.

Crossrefs

Cf. A000005 (tau_2), A007425 (tau_3), A007426 (tau_4), A061200 (tau_5).

Cf. A061204.

Column k=6 of A077592.

Programs

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 6], {n, 68}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

PARI

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] = binomial(f[i,2] + 5, f[i,2]); f[i,2]=1); factorback(f); \\ Michel Marcus, Jun 09 2014

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^6)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

Python

from math import prod, comb
from sympy import factorint
def A034695(n): return prod(comb(5+e,5) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

Formula

Dirichlet g.f.: zeta^6(s).

Multiplicative with a(p^e) = binomial(e+5, e). - Mitch Harris, Jun 27 2005

The Piltz divisor functions hold for tau_j(*)tau_k = tau_{j+k}, where (*) means Dirichlet convolution.

G.f.: Sum_{k>=1} tau_5(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Extensions

More terms from Robert G. Wilson v, Nov 02 2005