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

0,7

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

Links

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

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A306286 A226651 A073711 * A319179 A114653 A069481

Adjacent sequences: A071687 A071688 A071689 * A071691 A071692 A071693

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 23 2002

Extensions

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

Status

approved