A200214 Ordered factorizations of n with 3 distinct parts, all > 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 18, 0, 0, 0, 6, 0, 6, 0, 0, 0, 6, 0, 18, 0, 0, 0, 0, 0, 6, 0, 12, 0, 0, 0, 18

Offset

1,24

Links

Antti Karttunen, Table of n, a(n) for n = 1..1001

Benny Chor, Paul Lemke, Ziv Mador, On the number of ordered factorizations of natural numbers, Discrete Mathematics, Vol. 214[1], 2000, p. 123-133.

Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1), 2006.

Formula

a(n) = 6*A122180(n). - Antti Karttunen, Jul 08 2017

Example

a(24) = 6 = card({{2,3,4}, {2,4,3}, {3,2,4}, {3,4,2}, {4,2,3}, {4,3,2}}).
a(64) = 6 = card({{2,4,8}, {2,8,4}, {4,2,8}, {4,8,2}, {8,2,4}, {8,4,2}}).

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 && Length[# // Union] == 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 *)

Prog

(PARI) A200214(n) = { my(s=0); fordiv(n, x, if((x>1)&&(x<n), for(y=x+1, n-1, for(z=y+1, n-1, if(x*y*z==n, s++))))); (6*s); }; \\ (After A122180, still quite naive) - Antti Karttunen, Jul 09 2017

Crossrefs

Cf. A025487, A122180, A200213.

Sequence in context: A340979 A102638 A187794 * A294887 A331979 A005872

Adjacent sequences: A200211 A200212 A200213 * A200215 A200216 A200217

Keyword

nonn

Author

Peter Luschny, Nov 14 2011

Extensions

Description clarified, term a(0) removed and a second example added by Antti Karttunen, Jul 09 2017

Status

approved