A122180 Number of ways to write n as n = x*y*z with 1 < x < y < z < n.

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

Offset: 1

Author

Rick L. Shepherd, Aug 23 2006

Keywords

• nonn

Comments

x,y,z are distinct proper factors of n. See A122181 for n such that a(n) > 0.

If n has at most five divisors then a(n) = 0. - David A. Corneth, Oct 24 2024

Examples

			a(48) = 2 because 48 = 2*3*8 = 2*4*6, two products of three distinct proper factors of 48.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A034836, A088432, A088433, A088434, A122179, A122181, A200214.

Programs

PARI

for(n=1,105, t=0; for(x=2,n-1, for(y=x+1,n-1, for(z=y+1,n-1, if(x*y*z==n, t++)))); print1(t,", "))

PARI

A122180(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Jul 08 2017

PARI

a(n) = {
	my(d = divisors(n));
	if(#d <= 5, return(0));
	my(res = 0, q);
	for(i = 2, #d,
		q = d[#d + 1 - i];
		if(d[i]^2 > q,
			return(res)
		);
		for(j = i + 1, #d,
			qj = q/d[j];
			if(qj <= d[j],
				next(2)
			);
			if(denominator(qj) == 1 && n % qj == 0,
				res++
			);
		);
	);
	res
} \\ David A. Corneth, Oct 24 2024

Formula

a(n) = A200214(n)/6. - Antti Karttunen, Jul 08 2017