OFFSET
1,2
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Nov 02 2005
STATUS
approved