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

%I A071690 #24 Aug 02 2015 17:17:15
%S A071690 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,
%T A071690 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,
%U A071690 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
%N A071690 Number of ways to write n as n = x*y*z*t/(x+y+z+t) with 1 <= x <= y <= z <= t <= n.
%C A071690 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
%H A071690 David A. Corneth, <a href="/A071690/b071690.txt">Table of n, a(n) for n = 0..9999</a>
%e A071690 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
%o A071690 (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))))),","))
%o A071690 (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
%Y A071690 Cf. A065015 (the same without the constraint t <= n).
%K A071690 easy,nonn
%O A071690 0,7
%A A071690 _Benoit Cloitre_, Jun 23 2002
%E A071690 a(0)=0 and a(1)=0 prepended by _David A. Corneth_, Jul 31 2015