A200221 Ordered factorizations of n with 3 parts.
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, 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, 0, 3, 0, 6, 0, 27, 0, 0, 3, 3, 0, 6, 0, 18, 3, 0, 0, 21
Offset: 1
Keywords
Examples
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}}).
References
- Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(numtheory): b:= proc(n) option remember; expand((`if`(isprime(n), 0, add(b(n/d), d=divisors(n) minus {1, n}))+1)*x) end: a:= n-> coeff(b(n), x, 3): seq(a(n), n=1..100); # Alois P. Heinz, Dec 07 2014 -
Mathematica
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 *) nn = 200; f[list_, i_] := list[[i]]; a = Prepend[Table[1, {nn}], 0]; c = Table[DirichletConvolve[f[a, n], f[a, n], n, m], {m, 1, nn}]; Table[DirichletConvolve[f[a, n], f[c, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Apr 06 2020 *)
Formula
Dirichlet g.f.: (zeta(s)-1)^3. - Geoffrey Critzer, Apr 06 2020
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