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

%I A092575 #16 Feb 16 2025 08:32:52
%S A092575 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,
%T A092575 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,
%U A092575 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
%N A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.
%H A092575 Robert Israel, <a href="/A092575/b092575.txt">Table of n, a(n) for n = 1..10000</a>
%H A092575 E. Akhtarkavan, M. F. M. Salleh and O. Sidek, <a href="https://www.idosi.org/wasj/wasj21(2)13/2.pdf">Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000</a>, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
%H A092575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Eulers6nPlus1Theorem.html">Eulers 6n Plus 1 Theorem</a>
%p A092575 N:= 200: # to get a(1)..a(N)
%p A092575 V:= Vector(N):
%p A092575 for y from 1 to floor(sqrt(N/3-1)) do
%p A092575   for x from 1 to floor(sqrt(N-3*y^2)) do
%p A092575     if igcd(x,y) = 1 then V[x^2 + 3*y^2]:= V[x^2+3*y^2]+1
%p A092575     fi
%p A092575 od od:
%p A092575 convert(V,list); # _Robert Israel_, Apr 03 2017
%t A092575 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 *)
%Y A092575 Cf. A002476, A092572, A092573, A092574.
%K A092575 nonn
%O A092575 1,28
%A A092575 _Eric W. Weisstein_, Feb 28 2004