This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061200 #41 May 06 2025 03:31:58
%S A061200 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,
%T A061200 35,75,5,125,5,126,25,25,25,225,5,25,25,175,5,125,5,75,75,25,5,350,15,
%U A061200 75,25,75,5,175,25,175,25,25,5,375,5,25,75,210,25,125,5,75,25,125,5
%N A061200 tau_5(n) = number of ordered 5-factorizations of n.
%H A061200 Seiichi Manyama, <a href="/A061200/b061200.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique Pérez Herrero)
%F A061200 tau_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k=n}|, number of ordered k-factorizations of n.
%F A061200 tau_k(p^m) = (-1)^(k-1)*binomial(-m-1,k-1), p prime.
%F A061200 limit(tau_k(n)/n^epsilon, n=infinity) = 0, for any epsilon>0.
%F A061200 tau_k(n) = Sum_{d|n} tau_(k-1)(d), tau_1(n)=1.
%F A061200 Dirichlet g.f.: (zeta(s))^k.
%F A061200 For explicit formula, see A007425.
%F A061200 G.f.: Sum_{k>=1} tau_4(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%t A061200 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_ *)
%t A061200 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 *)
%o A061200 (PARI) for(n=1,100,print1(sumdiv(n,k,sumdiv(k,x,sumdiv(x,y,numdiv(y)))),","))
%o A061200 (PARI) a(n)=sumdivmult(n,k,sumdivmult(k,x,sumdivmult(x,y,numdiv(y)))) \\ _Charles R Greathouse IV_, Sep 09 2014
%o A061200 (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
%o A061200 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^5)[n]), ", ")) \\ _Vaclav Kotesovec_, May 06 2025
%o A061200 (Python)
%o A061200 from math import prod, comb
%o A061200 from sympy import factorint
%o A061200 def A061200(n): return prod(comb(4+e,4) for e in factorint(n).values()) # _Chai Wah Wu_, Dec 22 2024
%Y A061200 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 (partial sums), A061204.
%Y A061200 Column k=5 of A077592.
%K A061200 easy,nonn,mult
%O A061200 1,2
%A A061200 _Vladeta Jovovic_, Apr 21 2001