A088434 - OEIS

A088434

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

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, 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, 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

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OFFSET

1,12

COMMENTS

a(n)=0 iff n=1 or n prime or n prime^2: a(A000430(n)) = 0.

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

LINKS

Antti Karttunen and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen.)

Michael De Vlieger, Records and first positions of records in A088434.

EXAMPLE

n=12: (1,2,6), (1,3,4): therefore a(12)=2;

n=18: (1,2,9), (1,3,6): therefore a(18)=2.

MATHEMATICA

Table[Length[Select[Cases[Subsets[Divisors[n], {3}], {x_, y_, z_}->x*y*z], #==n &]], {n, 102}] (* Jayanta Basu, May 23 2013 *)

PROG

(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

CROSSREFS

Cf. A034836, A088432, A088433, A122179, A122180.