A111217 d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).

1, 7, 7, 28, 7, 49, 7, 84, 28, 49, 7, 196, 7, 49, 49, 210, 7, 196, 7, 196, 49, 49, 7, 588, 28, 49, 84, 196, 7, 343, 7, 462, 49, 49, 49, 784, 7, 49, 49, 588, 7, 343, 7, 196, 196, 49, 7, 1470, 28, 196, 49, 196, 7, 588, 49, 588, 49, 49, 7, 1372, 7, 49, 196, 924, 49, 343, 7, 196

Offset: 1

Author

Gerald McGarvey, Oct 25 2005

Keywords

• mult
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Crossrefs

Cf. tau_k(n) for k>=2: A000005, A007425, A007426, A061200, A034695, A111218 - A111221, A111306.

Column k=7 of A077592.

Programs

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 7], {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, 7], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

PARI

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,x,numdiv(x)))))),","))

PARI

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+6, 6)) \\ Charles R Greathouse IV, Oct 28 2017

Formula

Dirichlet convolution of A000012 with A034695, or of A000005 with A061200, or of A007425 with A007426. Dirichlet g.f. zeta^7(s). - R. J. Mathar, Apr 01 2011

G.f.: Sum_{k>=1} tau_6(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+6,6). - Amiram Eldar, Sep 13 2020