A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Author

Eric W. Weisstein, Feb 28 2004

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
• Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Crossrefs

Cf. A002476, A092572, A092573, A092574.

Programs

Maple

N:= 200: # to get a(1)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  for x from 1 to floor(sqrt(N-3*y^2)) do
    if igcd(x,y) = 1 then V[x^2 + 3*y^2]:= V[x^2+3*y^2]+1
    fi
od od:
convert(V,list); # Robert Israel, Apr 03 2017

Mathematica

r[n_] := Reduce[ x > 0 && y > 0 && GCD[x, y] == 1 && n == x^2 + 3 y^2, {x, y}, Integers]; a[n_] := Which[ r[n] === False, 0, r[n][[0]] === And, 1, True, Length[r[n]]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 31 2012 *)