A071690 Number of ways to write n as n = x*y*z*t/(x+y+z+t) with 1 <= x <= y <= z <= t <= n.

0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 6, 1, 8, 1, 7, 6, 9, 1, 15, 1, 10, 5, 8, 2, 21, 5, 10, 6, 15, 2, 28, 1, 16, 8, 11, 11, 27, 2, 9, 6, 30, 1, 35, 3, 18, 17, 9, 3, 43, 8, 30, 8, 17, 2, 51, 8, 32, 7, 14, 2, 55, 2, 10, 20, 37, 14, 40, 2, 19, 10, 50, 2, 65, 2, 13, 31, 21, 9, 46, 2, 57, 19, 15, 3, 67, 11, 11, 9, 41, 2, 89, 15, 22, 12

Offset: 0

Author

Benoit Cloitre, Jun 23 2002

Keywords

• easy
• nonn

Comments

Using x <= y <= z <= t, we have n = xyzt/(x+y+z+t) >= xyzt/(4t), i.e., 4n >= xyz >= xy^2 >= x^3, whence z <= 4n/(xy), y^2 <= 4n/x, and x^3 <= 4n. We also must have n*(x+y+z+t) = xyzt <=> n*(x+y+z) = (xyz-n)*t, i.e., xyz > n and z <= t = n*(x+y+z)/(xyz-n) <= n <=> x+y+z <= xyz-n <=> x+y+n <= (xy-1)z, whence y > 1/x and z >= (x+y+n)/(xy-1). - M. F. Hasler, Aug 02 2015

Examples

			As (x, y, z, t) = (2, 2, 2, 2) is the only quadruple that gives n = 2, a(2) = 1. - _David A. Corneth_, Jul 31 2015
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A065015 (the same without the constraint t <= n).

Programs

PARI

for(n=1,50,print1(sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,if(a*b*c*d/(a+b+c+d)-n,0,1))))),","))

PARI

A071690(n,d=0)=sum(x=1,sqrtn(4*n+.5,3), sum(y=max(x,2), sqrtint(4*n\x), sum(z=max(y,ceil((x+y+n)/(x*y-1))), 4*n\(x*y), (x+y+z)*n%(x*y*z-n)==0&&(x+y+z)*n>=(x*y*z-n)*z&&!(d&&print1([x,y,z,t=(x+y+z)*n/(x*y*z-n),x*y*z*t/(x+y+z+t)])))))  \\ M. F. Hasler, Aug 02 2015

Extensions

a(0)=0 and a(1)=0 prepended by David A. Corneth, Jul 31 2015