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
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