A111220 d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).

1, 10, 10, 55, 10, 100, 10, 220, 55, 100, 10, 550, 10, 100, 100, 715, 10, 550, 10, 550, 100, 100, 10, 2200, 55, 100, 220, 550, 10, 1000, 10, 2002, 100, 100, 100, 3025, 10, 100, 100, 2200, 10, 1000, 10, 550, 550, 100, 10, 7150, 55, 550, 100, 550, 10, 2200, 100

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_9(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+9,9). - 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, 10], {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, 10], {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, x, numdiv(x))))))))), ", "))
(PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+9, 9)) \\ Charles R Greathouse IV, Oct 28 2017

Crossrefs

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

Column k=10 of A077592.

Sequence in context: A341836 A328530 A238017 * A341253 A106789 A270012

Adjacent sequences: A111217 A111218 A111219 * A111221 A111222 A111223

Keyword

mult,nonn

Author

Gerald McGarvey, Oct 25 2005

Status

approved