A111218 d_8(n), tau_8(n), number of ordered factorizations of n as n = rstuvwxy (8-factorizations).

1, 8, 8, 36, 8, 64, 8, 120, 36, 64, 8, 288, 8, 64, 64, 330, 8, 288, 8, 288, 64, 64, 8, 960, 36, 64, 120, 288, 8, 512, 8, 792, 64, 64, 64, 1296, 8, 64, 64, 960, 8, 512, 8, 288, 288, 64, 8, 2640, 36, 288, 64, 288, 8, 960, 64, 960, 64, 64, 8, 2304, 8, 64, 288, 1716, 64, 512, 8

Offset: 1

Author

Gerald McGarvey, Oct 25 2005

Keywords

• mult
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=8 of A077592.

Programs

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 8], {n, 67}] (* 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, 8], {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,m,sumdiv(m,x,numdiv(x))))))),","))

PARI

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+7, 7)) \\ Charles R Greathouse IV, Oct 28 2017

Formula

G.f.: Sum_{k>=1} tau_7(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+7,7). - Amiram Eldar, Sep 13 2020