%I #51 Sep 13 2020 02:57:07
%S 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 21,36,56,126,6,216,6,252,36,36,36,441,6,36,36,336,6,216,6,126,126,36,
%U 6,756,21,126,36,126,6,336,36,336,36,36,6,756,6,36,126,462,36,216,6,126
%N 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 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 29 and 38
%D Leveque, William J., Fundamentals of Number Theory. New York:Dover Publications, 1996, ISBN 9780486689067, p .167-Exercise 5.b.
%H 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 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 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 Dirichlet g.f.: zeta^6(s).
%F Multiplicative with a(p^e) = binomial(e+5, e). - _Mitch Harris_, Jun 27 2005
%F The Piltz divisor functions hold for tau_j(*)tau_k = tau_{j+k}, where (*) means Dirichlet convolution.
%F G.f.: Sum_{k>=1} tau_5(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%t 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 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 (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
%Y Cf. A000005 (tau_2), A007425 (tau_3), A007426 (tau_4), A061200 (tau_5).
%Y Cf. A061204.
%Y Column k=6 of A077592.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Nov 02 2005