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%I #20 Aug 25 2017 23:47:32
%S 0,0,0,0,0,1,0,1,0,1,0,2,0,1,1,1,0,2,0,2,1,1,0,4,0,1,1,2,0,4,0,2,1,1,
%T 1,4,0,1,1,4,0,4,0,2,2,1,0,6,0,2,1,2,0,4,1,4,1,1,0,8,0,1,2,3,1,4,0,2,
%U 1,4,0,8,0,1,2,2,1,4,0,6,1,1,0,8,1,1,1,4,0,8,1,2,1,1,1,9,0,2,2,4,0,4,0,4,4,1,0,8,0,4,1,6,0,4,1,2,2,1,1,14
%N Number of ways to write n as n = u*v*w with 1 <= u < v < w.
%C a(n)=0 iff n=1 or n prime or n prime^2: a(A000430(n)) = 0.
%C The integers a(n)+1 equal A045778(n) for n < 120 and differ at all n that admit factorization into 4 or more distinct factors, the smallest ones being n = 120 = 2*3*4*5, n = 144 = 2*3*4*6, n = 168 = 2*3*4*7, n = 180 = 2*3*5*6, ..., later continuing n = 312 = 2*3*4*13, n = 320 = 2*4*5*8, n = 324 = 2*3*6*9, n = 330 = 2*3*5*11, ... Coincidentally, A068350(5) to A068350(19) start this list. - _R. J. Mathar_, Jul 19 2007
%H Antti Karttunen and Michael De Vlieger, <a href="/A088434/b088434.txt">Table of n, a(n) for n = 1..10000</a> (first 2048 terms from Antti Karttunen.)
%H Michael De Vlieger, <a href="/A088434/a088434.txt">Records and first positions of records in A088434.</a>
%e n=12: (1,2,6), (1,3,4): therefore a(12)=2;
%e n=18: (1,2,9), (1,3,6): therefore a(18)=2.
%t Table[Length[Select[Cases[Subsets[Divisors[n],{3}],{x_,y_,z_}->x*y*z],#==n &]],{n,102}] (* _Jayanta Basu_, May 23 2013 *)
%o (PARI) A088434(n) = { my(s=0); fordiv(n, u, for(v=u+1, n-1, for(w=v+1, n, if(u*v*w==n, s++)))); (s); }; \\ _Antti Karttunen_, Aug 24 2017
%Y Cf. A034836, A088432, A088433, A122179, A122180.
%K nonn
%O 1,12
%A _Reinhard Zumkeller_, Oct 01 2003
%E Data section extended to 120 terms by _Antti Karttunen_, Aug 24 2017
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