

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
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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 Nd 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 <= Md, 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] (* JeanFrançois 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, b1, n=a+b+c+d; if(n<=m, if((a/bc/d)==((ac)/(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



