This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A034695 #56 May 06 2025 03:26:37
%S A034695 1,6,6,21,6,36,6,56,21,36,6,126,6,36,36,126,6,126,6,126,36,36,6,336,
%T A034695 21,36,56,126,6,216,6,252,36,36,36,441,6,36,36,336,6,216,6,126,126,36,
%U A034695 6,756,21,126,36,126,6,336,36,336,36,36,6,756,6,36,126,462,36,216,6,126
%N 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.
%D A034695 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 29 and 38
%D A034695 Leveque, William J., Fundamentals of Number Theory. New York:Dover Publications, 1996, ISBN 9780486689067, p .167-Exercise 5.b.
%H A034695 Seiichi Manyama, <a href="/A034695/b034695.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique Pérez Herrero)
%H A034695 E. Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2009/08/splitting-fta-functions-ii.html">Piltz Divisor functions (1)</a>, Psychedelic Geometry Blogspot, Dec 21 2009.
%H A034695 E. Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2009/08/splitting-fta-functions-iii.html">Piltz Divisor functions (2)</a>, Psychedelic Geometry Blogspot, Dec 24 2009.
%F A034695 Dirichlet g.f.: zeta^6(s).
%F A034695 Multiplicative with a(p^e) = binomial(e+5, e). - _Mitch Harris_, Jun 27 2005
%F A034695 The Piltz divisor functions hold for tau_j(*)tau_k = tau_{j+k}, where (*) means Dirichlet convolution.
%F A034695 G.f.: Sum_{k>=1} tau_5(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%t A034695 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 *)
%t A034695 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 *)
%o A034695 (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
%o A034695 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^6)[n]), ", ")) \\ _Vaclav Kotesovec_, May 06 2025
%o A034695 (Python)
%o A034695 from math import prod, comb
%o A034695 from sympy import factorint
%o A034695 def A034695(n): return prod(comb(5+e,5) for e in factorint(n).values()) # _Chai Wah Wu_, Dec 22 2024
%Y A034695 Cf. A000005 (tau_2), A007425 (tau_3), A007426 (tau_4), A061200 (tau_5).
%Y A034695 Cf. A061204.
%Y A034695 Column k=6 of A077592.
%K A034695 nonn,mult
%O A034695 1,2
%A A034695 _N. J. A. Sloane_
%E A034695 More terms from _Robert G. Wilson v_, Nov 02 2005