A122180 - OEIS

%I #15 Oct 25 2024 09:36:14

%S 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,

%T 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,

%U 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

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

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

%C If n has at most five divisors then a(n) = 0. - _David A. Corneth_, Oct 24 2024

%H David A. Corneth, <a href="/A122180/b122180.txt">Table of n, a(n) for n = 1..10000</a> (first 1001 terms from Antti Karttunen)

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A200214(n)/6. - _Antti Karttunen_, Jul 08 2017

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

%o (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,", "))

%o (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

%o (PARI) a(n) = {

%o 	my(d = divisors(n));

%o 	if(#d <= 5, return(0));

%o 	my(res = 0, q);

%o 	for(i = 2, #d,

%o 		q = d[#d + 1 - i];

%o 		if(d[i]^2 > q,

%o 			return(res)

%o 		);

%o 		for(j = i + 1, #d,

%o 			qj = q/d[j];

%o 			if(qj <= d[j],

%o 				next(2)

%o 			);

%o 			if(denominator(qj) == 1 && n % qj == 0,

%o 				res++

%o 			);

%o 		);

%o 	);

%o 	res

%o } \\ _David A. Corneth_, Oct 24 2024

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

%K nonn

%O 1,48

%A _Rick L. Shepherd_, Aug 23 2006