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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
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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

Sequence in context: A263412 A321258 A331510 * A124035 A204184 A157897

Adjacent sequences:  A319851 A319852 A319853 * A319855 A319856 A319857

KEYWORD

nonn,look

AUTHOR

Hugo Pfoertner and Rainer Rosenthal, Sep 29 2018

STATUS

approved

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Last modified June 19 03:31 EDT 2021. Contains 345125 sequences. (Running on oeis4.)