%I
%S 0,0,0,0,0,0,0,1,0,0,1,0,1,2,0,1,1,3,1,1,1,3,3,2,1,3,2,4,2,2,3,6,3,4,
%T 2,5,3,7,4,3,4,6,5,9,2,7,4,6,5,9,5,6,6,8,3,9,7,12,7,6,6,11,5,12,6,11,
%U 7,12,6,9,10,12,7,16,5,13,8,14,9,11,9,15,11,14
%N Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a/b  c/d = (a  c)/(b + d).
%C Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a*d^2 = b^2*c.  _Robert Israel_, Oct 04 2018
%H Robert Israel, <a href="/A319854/b319854.txt">Table of n, a(n) for n = 1..10000</a> (first 500 terms from Hugo Pfoertner)
%e a(8) = 1: 4/2  1/1 = (4  1)/(2 + 1) = 1;
%e a(11) = 1: 4/4  1/2 = (4  1)/(4 + 2) = 1/2;
%e a(13) = 1: 8/2  2/1 = (8  2)/(2 + 1) = 2;
%e a(14) = 2: 4/6  1/3 = (4  1)/(6 + 3) = 1/3, 9/3  1/1 = (9  1)/(3 + 1) = 2;
%e a(16) = 1: 8/4  2/2 = (8  2)/(4 + 2) = 1;
%e a(17) = 1: 4/8  1/4 = (4  1)/(8 + 4) = 1/4;
%e a(18) = 3: 9/3  4/2 = (9  4)/(3 + 2) = 1, 9/6  1/2 = (9  1)/(6 + 2) = 1, 12/2  3/1 = (12  3)/(2 + 1) = 3.
%p N:= 1000: # for a(1)..a(N)
%p V:= Vector(N):
%p for d from 1 to N/2 do
%p for b from d+1 to Nd do
%p u:= d^2/igcd(b,d)^2;
%p for c from u by u do
%p v:= c*b^2/d^2+b+c+d;
%p if v > N then break fi;
%p V[v]:= V[v]+1
%p od od od:
%p convert(V,list); # _Robert Israel_, Oct 04 2018
%t M = 100; Clear[V]; V[_] = 0;
%t For[d = 1, d <= M/2, d++,
%t For[b = d+1, b <= Md, b++,
%t u = d^2/GCD[b, d]^2;
%t For[c = u, True, c = c+u,
%t v = c*b^2/d^2 + b + c + d;
%t If[v > M, Break[]];
%t V[v] = V[v]+1
%t ]]];
%t Array[V, M] (* _JeanFrançois Alcover_, Apr 02 2019, after _Robert Israel_ *)
%o (PARI) m=84;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,b1,n=a+b+c+d;if(n<=m,if((a/bc/d)==((ac)/(b+d)),v[n]++))))));v
%Y Cf. A026810, A319853, A320311, A320312, A320313.
%K nonn,look
%O 1,14
%A _Hugo Pfoertner_ and _Rainer Rosenthal_, Sep 29 2018
