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A088432 Number of ways to write n as n = u*v*w with 1 <= u < v <= w.

%I A088432 #26 Dec 17 2021 20:37:03
%S A088432 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,2,0,3,0,2,1,1,0,4,1,1,1,2,0,4,0,3,1,1,
%T A088432 1,5,0,1,1,4,0,4,0,2,2,1,0,7,1,3,1,2,0,4,1,4,1,1,0,8,0,1,2,4,1,4,0,2,
%U A088432 1,4,0,9,0,1,3,2,1,4,0,6,2,1,0,8,1,1,1,4,0,8,1,2,1,1,1,9,0,3,2,6,0,4,0,4,4,1,0,9,0,4,1,6,0,4,1,2,2,1,1,14
%N A088432 Number of ways to write n as n = u*v*w with 1 <= u < v <= w.
%H A088432 Antti Karttunen, <a href="/A088432/b088432.txt">Table of n, a(n) for n = 1..3003</a>
%F A088432 a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0, a(A002808(n)) > 0.
%F A088432 a(n) = 1 iff n has 3 or 4 divisors (A323644) (see examples). - _Bernard Schott_, Dec 13 2021
%F A088432 a(n) = 2 if n = p^2*q, p<q primes (A096156) or n = p^4 (A030514) (see examples). - _Bernard Schott_, Dec 16 2021
%e A088432 n=12: (1,2,6), (1,3,4): therefore a(12)=2;
%e A088432 n=18: (1,2,9), (1,3,6), (2,3,3): therefore a(18)=3.
%e A088432 For n = p*q, p < q primes:  n = 1 * p * q, so a(n) = 1.
%e A088432 For n = p^2, p prime: n = 1 * p * p, so a(n) = 1.
%e A088432 For n = p^3, p prime: n = 1 * p * p^2, so a(n) = 1.
%e A088432 For n = p*q^2, p < q < p^2: n = 1 * p * pq = 1* q * p^2, so a(n) = 2 (see n=12).
%e A088432 For n = p*q^2, p < p^2 < q: n = 1 * p * pq = 1 * p^2 * q, so a(n) = 2
%e A088432 For n = p^4, p prime: n = 1 * p * p^3 = 1 * p^2 * p^2, so a(n) = 2.
%t A088432 a[n_] := Module[{s = 0}, Do[Do[Do[If[u v w == n, s++], {w, v, n}], {v, u + 1, n - 1}], {u, Divisors[n]}]; s];
%t A088432 Array[a, 120] (* _Jean-François Alcover_, Dec 10 2021, after _Antti Karttunen_ *)
%o A088432 (PARI) A088432(n) = { my(s=0); fordiv(n, u, for(v=u+1, n-1, for(w=v, n, if(u*v*w==n, s++)))); (s); }; \\ _Antti Karttunen_, Aug 24 2017
%Y A088432 Cf. A034836, A088433, A088434, A122179, A122180, A323644.
%K A088432 nonn
%O A088432 1,12
%A A088432 _Reinhard Zumkeller_, Oct 01 2003
%E A088432 Data section extended to 120 terms by _Antti Karttunen_, Aug 24 2017