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

%I A200214 #49 Oct 24 2024 03:51:59
%S A200214 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,
%T A200214 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,
%U A200214 0,0,6,0,18,0,0,0,0,0,6,0,12,0,0,0,18
%N A200214 Ordered factorizations of n with 3 distinct parts, all > 1.
%H A200214 Antti Karttunen, <a href="/A200214/b200214.txt">Table of n, a(n) for n = 1..1001</a>
%H A200214 Benny Chor, Paul Lemke, Ziv Mador, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00223-X">On the number of ordered factorizations of natural numbers</a>, Discrete Mathematics, Vol. 214[1], 2000, p. 123-133.
%H A200214 Arnold Knopfmacher and Michael Mays, <a href="https://www.mathematica-journal.com/issue/v10i1/contents/Factorizations/Factorizations_3.html">Ordered and Unordered Factorizations of Integers</a>, The Mathematica Journal, Vol 10 (1), 2006.
%F A200214 a(n) = 6*A122180(n). - _Antti Karttunen_, Jul 08 2017
%e A200214 a(24) = 6 = card({{2,3,4}, {2,4,3}, {3,2,4}, {3,4,2}, {4,2,3}, {4,3,2}}).
%e A200214 a(64) = 6 = card({{2,4,8}, {2,8,4}, {4,2,8}, {4,8,2}, {8,2,4}, {8,4,2}}).
%t A200214 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 *)
%o A200214 (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
%Y A200214 Cf. A025487, A122180, A200213.
%K A200214 nonn
%O A200214 1,24
%A A200214 _Peter Luschny_, Nov 14 2011
%E A200214 Description clarified, term a(0) removed and a second example added by _Antti Karttunen_, Jul 09 2017