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 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). 5

%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 N-d 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 <= M-d, 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] (* _Jean-François 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,b-1,n=a+b+c+d;if(n<=m,if((a/b-c/d)==((a-c)/(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

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Last modified July 30 23:12 EDT 2021. Contains 346365 sequences. (Running on oeis4.)