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

1,48

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

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

Formula

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

Example

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

Prog

(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)&&(x<n), for(y=x+1, n-1, for(z=y+1, n-1, if(x*y*z==n, s++))))); (s); }; \\ Just slightly optimized from the above. - Antti 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

Crossrefs

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

Sequence in context: A070141 A088722 A336917 * A363742 A378446 A033772

Adjacent sequences: A122177 A122178 A122179 * A122181 A122182 A122183

Keyword

nonn

Author

Rick L. Shepherd, Aug 23 2006

Status

approved