A319854 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).

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, 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, 7, 12, 6, 9, 10, 12, 7, 16, 5, 13, 8, 14, 9, 11, 9, 15, 11, 14

Offset: 1

Author

Hugo Pfoertner and Rainer Rosenthal, Sep 29 2018

Keywords

• nonn
• look

Comments

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

Examples

			a(8) = 1: 4/2 - 1/1 = (4 - 1)/(2 + 1) = 1;
a(11) = 1: 4/4 - 1/2 = (4 - 1)/(4 + 2) = 1/2;
a(13) = 1: 8/2 - 2/1 = (8 - 2)/(2 + 1) = 2;
a(14) = 2: 4/6 - 1/3 = (4 - 1)/(6 + 3) = 1/3, 9/3 - 1/1 = (9 - 1)/(3 + 1) = 2;
a(16) = 1: 8/4 - 2/2 = (8 - 2)/(4 + 2) = 1;
a(17) = 1: 4/8 - 1/4 = (4 - 1)/(8 + 4) = 1/4;
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 500 terms from Hugo Pfoertner)

Crossrefs

Cf. A026810, A319853, A320311, A320312, A320313.

Programs

Maple

N:= 1000: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N/2 do
  for b from d+1 to N-d do
    u:= d^2/igcd(b,d)^2;
    for c from u by u  do
      v:= c*b^2/d^2+b+c+d;
      if v > N then break fi;
      V[v]:= V[v]+1
od od od:
convert(V,list); # Robert Israel, Oct 04 2018

Mathematica

M = 100; Clear[V]; V[_] = 0;
For[d = 1, d <= M/2, d++,
  For[b = d+1, b <= M-d, b++,
    u = d^2/GCD[b, d]^2;
    For[c = u, True, c = c+u,
      v = c*b^2/d^2 + b + c + d;
      If[v > M, Break[]];
      V[v] = V[v]+1
]]];
Array[V, M] (* Jean-François Alcover, Apr 02 2019, after Robert Israel *)

PARI

m=84;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,b-1,n=a+b+c+d;if(n<=m,if((a/b-c/d)==((a-c)/(b+d)),v[n]++))))));v