A111221 d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

1, 11, 11, 66, 11, 121, 11, 286, 66, 121, 11, 726, 11, 121, 121, 1001, 11, 726, 11, 726, 121, 121, 11, 3146, 66, 121, 286, 726, 11, 1331, 11, 3003, 121, 121, 121, 4356, 11, 121, 121, 3146, 11, 1331, 11, 726, 726, 121, 11, 11011, 66, 726, 121, 726, 11, 3146, 121

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)
• Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

Crossrefs

Cf. tau_1(n): A000012

Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.

Cf. tau_7(n)...tau_10(n): A111217, A111218, A111219, A111220.

Cf. tau_12(n): A111306.

Column k=11 of A077592.

Programs

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 11], {n, 55}] (* 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, 11], {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,p,sumdiv(p,q,sumdiv(q,x,numdiv(x)))))))))),","))

PARI

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+10, 10)) \\ Charles R Greathouse IV, Oct 28 2017

Formula

G.f.: Sum_{k>=1} tau_10(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+10,10). - Amiram Eldar, Sep 13 2020