OFFSET
1,14
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
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000 (first 500 terms from Hugo Pfoertner)
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Hugo Pfoertner and Rainer Rosenthal, Sep 29 2018
STATUS
approved