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

%I A122180 #15 Oct 25 2024 09:36:14
%S A122180 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 A122180 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 A122180 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 A122180 Number of ways to write n as n = x*y*z with 1 < x < y < z < n.
%C A122180 x,y,z are distinct proper factors of n. See A122181 for n such that a(n) > 0.
%C A122180 If n has at most five divisors then a(n) = 0. - _David A. Corneth_, Oct 24 2024
%H A122180 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 A122180 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A122180 a(n) = A200214(n)/6. - _Antti Karttunen_, Jul 08 2017
%e A122180 a(48) = 2 because 48 = 2*3*8 = 2*4*6, two products of three distinct proper factors of 48.
%o A122180 (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 A122180 (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 A122180 (PARI) a(n) = {
%o A122180 	my(d = divisors(n));
%o A122180 	if(#d <= 5, return(0));
%o A122180 	my(res = 0, q);
%o A122180 	for(i = 2, #d,
%o A122180 		q = d[#d + 1 - i];
%o A122180 		if(d[i]^2 > q,
%o A122180 			return(res)
%o A122180 		);
%o A122180 		for(j = i + 1, #d,
%o A122180 			qj = q/d[j];
%o A122180 			if(qj <= d[j],
%o A122180 				next(2)
%o A122180 			);
%o A122180 			if(denominator(qj) == 1 && n % qj == 0,
%o A122180 				res++
%o A122180 			);
%o A122180 		);
%o A122180 	);
%o A122180 	res
%o A122180 } \\ _David A. Corneth_, Oct 24 2024
%Y A122180 Cf. A034836, A088432, A088433, A088434, A122179, A122181, A200214.
%K A122180 nonn
%O A122180 1,48
%A A122180 _Rick L. Shepherd_, Aug 23 2006