A122179 Number of ways to write n as n = x*y*z with 1<x<=y<=z<n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 3, 0, 1, 0, 1, 0, 1, 0, 6, 0, 0, 1, 1, 0, 1, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 6, 0, 1, 1, 3, 0, 1, 0, 2, 1

Offset

1,24

Comments

x,y,z are proper factors of n. a(n) > 0 iff n is a term of A033942; a(n) = 0 iff n is a term of A037143.

By Burnside on S3: nondecreasing triples with x*y*z = n give (A007425(n) + 3*A046951(n) + 2*A010057(n))/6; subtract A038548(n) to exclude triples containing 1. - Felix Huber, May 27 2026

Links

Felix Huber, Table of n, a(n) for n = 1..10000 (terms 1..2048 from Antti Karttunen, terms 2049..8192 from Seiichi Manyama)

Wikipedia, Burnside's lemma

Formula

a(n) = (A007425(n) + 3*A046951(n) + 2*A010057(n))/6 - A038548(n) = A034836(n) - A038548(n). - Felix Huber, May 27 2026

Example

a(24) = 2 because 24 = 2*2*6 = 2*3*4, two products of three proper factors of 24.

Maple

A122179 := proc(n)
    local i, l, x;
    l := [seq(i[2], i in ifactors(n)[2])];
    (mul(binomial(x + 2, 2), x in l) + 3*mul(floor(x/2) + 1, x in l) + 2*`if`(andmap(x -> x mod 3 = 0, l), 1, 0))/6 - ceil(mul(x + 1, x in l)/2);
end proc:
seq(A122179(n), n = 1 .. 88); # Felix Huber, May 27 2026

Prog

(PARI) for(n=1, 105, t=0; for(x=2, n-1, for(y=x, n-1, for(z=y, n-1, if(x*y*z==n, t++)))); print1(t, ", "))
(PARI) A122179(n) = { my(s=0); fordiv(n, x, if((x>1)&&(x<n), for(y=x, n-1, for(z=y, n-1, if(x*y*z==n, s++))))); (s); }; \\ Antti Karttunen, Aug 24 2017
(Python)
from math import prod, comb
from sympy import factorint
def A122179(n):
    f = factorint(n).values()
    return (prod(comb(e+2, 2) for e in f)+3*prod((e>>1)+1 for e in f)+((not any(e%3 for e in f))<<1))//6-(prod(e+1 for e in f)+1>>1) # Chai Wah Wu, May 28 2026

Crossrefs

Cf. A034836, A033942, A037143, A088432, A088433, A088434, A122180.

Cf. A007425, A010057, A038548, A046951.

Sequence in context: A378149 A309367 A347709 * A335877 A125203 A023565

Adjacent sequences: A122176 A122177 A122178 * A122180 A122181 A122182

Keyword

nonn

Author

Rick L. Shepherd, Aug 23 2006

Status

approved