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

1,2

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_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

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 *)

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, 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

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.

Sequence in context: A228768 A003876 A014461 * A241870 A243127 A088761

Adjacent sequences: A111218 A111219 A111220 * A111222 A111223 A111224

Keyword

mult,nonn

Author

Gerald McGarvey, Oct 25 2005

Status

approved