A061200 tau_5(n) = number of ordered 5-factorizations of n.

1, 5, 5, 15, 5, 25, 5, 35, 15, 25, 5, 75, 5, 25, 25, 70, 5, 75, 5, 75, 25, 25, 5, 175, 15, 25, 35, 75, 5, 125, 5, 126, 25, 25, 25, 225, 5, 25, 25, 175, 5, 125, 5, 75, 75, 25, 5, 350, 15, 75, 25, 75, 5, 175, 25, 175, 25, 25, 5, 375, 5, 25, 75, 210, 25, 125, 5, 75, 25, 125, 5

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Formula

tau_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k=n}|, number of ordered k-factorizations of n.

tau_k(p^m) = (-1)^(k-1)*binomial(-m-1,k-1), p prime.

limit(tau_k(n)/n^epsilon, n=infinity) = 0, for any epsilon>0.

tau_k(n) = Sum_{d|n} tau_(k-1)(d), tau_1(n)=1.

Dirichlet g.f.: (zeta(s))^k.

For explicit formula, see A007425.

G.f.: Sum_{k>=1} tau_4(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 5], {n, 77}] (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 5], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

Prog

(PARI) for(n=1, 100, print1(sumdiv(n, k, sumdiv(k, x, sumdiv(x, y, numdiv(y)))), ", "))
(PARI) a(n)=sumdivmult(n, k, sumdivmult(k, x, sumdivmult(x, y, numdiv(y)))) \\ Charles R Greathouse IV, Sep 09 2014
(PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+4, 4)) \\ Charles R Greathouse IV, Oct 28 2017

Crossrefs

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_6(n): A034695, (unordered) 2-factorization of n: A038548, (unordered) 3-factorization of n: A034836, A001055, A006218, A061201, A061202, A061203, A061204.

Column k=5 of A077592.

Sequence in context: A304300 A321653 A247940 * A255304 A339337 A330567

Adjacent sequences: A061197 A061198 A061199 * A061201 A061202 A061203

Keyword

easy,nonn,mult

Author

Vladeta Jovovic, Apr 21 2001

Status

approved