A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

1, 12, 12, 78, 12, 144, 12, 364, 78, 144, 12, 936, 12, 144, 144, 1365, 12, 936, 12, 936, 144, 144, 12, 4368, 78, 144, 364, 936, 12, 1728, 12, 4368, 144, 144, 144, 6084, 12, 144, 144, 4368, 12, 1728, 12, 936, 936, 144, 12, 16380, 78, 936, 144, 936, 12, 4368, 144

Offset

1,2

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.

Formula

G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+11,11). - Amiram Eldar, Sep 13 2020

Maple

b:= proc(n, k) option remember; `if`(k=1, 1,
      add(b(d, k-1), d=numtheory[divisors](n)))
    end:
a:= n-> b(n, 12):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 12 2024

Mathematica

tau[k_, 1]:=1; tau[k_, n_]:=Times@@(Binomial[#+k-1, k-1]&/@FactorInteger[n][[All, 2]]); Table[tau[12, n], {n, 1000}] (* Enrique Pérez Herrero, Jan 17 2013 *)

Prog

(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, r, sumdiv(r, x, numdiv(x))))))))))), ", "))
(PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+11, 11)) \\ Charles R Greathouse IV, Oct 28 2017

Crossrefs

Cf. tau_2(n)...tau_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.

Column k=12 of A077592.

Sequence in context: A003877 A161196 A328531 * A151777 A143478 A219400

Adjacent sequences: A111303 A111304 A111305 * A111307 A111308 A111309

Keyword

mult,nonn

Author

Gerald McGarvey, Nov 02 2005

Status

approved