cp's OEIS Frontend

A200221 Ordered factorizations of n with 3 parts.

%I A200221 #29 Apr 11 2020 07:09:40
%S A200221 0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,3,0,3,0,3,0,0,0,9,0,0,1,3,0,6,0,6,0,0,
%T A200221 0,12,0,0,0,9,0,6,0,3,3,0,0,18,0,3,0,3,0,9,0,9,0,0,0,21,0,0,3,10,0,6,
%U A200221 0,3,0,6,0,27,0,0,3,3,0,6,0,18,3,0,0,21
%N A200221 Ordered factorizations of n with 3 parts.
%D A200221 Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).
%H A200221 Alois P. Heinz, <a href="/A200221/b200221.txt">Table of n, a(n) for n = 1..10000</a>
%F A200221 Dirichlet g.f.: (zeta(s)-1)^3. - _Geoffrey Critzer_, Apr 06 2020
%F A200221 Sum_{k=1..n} a(k) ~ n*(log(n)^2/2 + (3*gamma - 4)*log(n) + 3*gamma^2 - 9*gamma - 3*sg1 + 7), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Apr 07 2020
%e A200221 a(24) = 9 = card({{4,3,2}, {4,2,3}, {3,4,2}, {3,2,4}, {2,4,3}, {2,3,4}, {6,2,2},{2,6,2}, {2,2,6}}).
%p A200221 with(numtheory):
%p A200221 b:= proc(n) option remember; expand((`if`(isprime(n), 0,
%p A200221       add(b(n/d), d=divisors(n) minus {1, n}))+1)*x)
%p A200221     end:
%p A200221 a:= n-> coeff(b(n), x, 3):
%p A200221 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 07 2014
%t A200221 OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of3 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 3 &] // Union}, Length[Permutations /@ of3 // Flatten[#, 1] &]]; Table[a[n], {n, 1, 84}] (* _Jean-François Alcover_, Jul 02 2013, copied and adapted from The Mathematica Journal *)
%t A200221 nn = 200; f[list_, i_] := list[[i]]; a = Prepend[Table[1, {nn}], 0];
%t A200221 c = Table[DirichletConvolve[f[a, n], f[a, n], n, m], {m, 1, nn}];
%t A200221 Table[DirichletConvolve[f[a, n], f[c, n], n, m], {m, 1, nn}] (* _Geoffrey Critzer_, Apr 06 2020 *)
%Y A200221 Cf. A200214.
%Y A200221 Column k=3 of A251683.
%K A200221 nonn
%O A200221 1,12
%A A200221 _Peter Luschny_, Nov 14 2011