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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-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 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.)