A111219 d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).

1, 9, 9, 45, 9, 81, 9, 165, 45, 81, 9, 405, 9, 81, 81, 495, 9, 405, 9, 405, 81, 81, 9, 1485, 45, 81, 165, 405, 9, 729, 9, 1287, 81, 81, 81, 2025, 9, 81, 81, 1485, 9, 729, 9, 405, 405, 81, 9, 4455, 45, 405, 81, 405, 9, 1485, 81, 1485, 81, 81, 9, 3645, 9, 81, 405, 3003, 81

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_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.

Column k=9 of A077592.

Programs

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 9], {n, 65}] (* 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, 9], {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,o,sumdiv(o,x,numdiv(x)))))))),","))

PARI

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+8, 8)) \\ Charles R Greathouse IV, Oct 28 2017

Formula

G.f.: Sum_{k>=1} tau_8(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+8,8). - Amiram Eldar, Sep 13 2020