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A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers.

%I A301452 #14 Mar 16 2025 07:30:13
%S A301452 0,2,2,2,2,5,3,2,4,4,4,5,5,5,6,2,4,6,5,4,9,5,4,5,5,7,10,5,6,7,8,2,6,6,
%T A301452 7,6,9,7,10,4,6,12,3,5,10,5,6,5,5,8,9,7,7,12,5,5,13,9,6,7,8,10,13,2,6,
%U A301452 8,10,6,15,9,9,6,10,9,12,7,8,13,6,4
%N A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers.
%C A301452 Conjecture: a(n) > 0 for all n > 1.
%C A301452 We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1.
%C A301452 We have verified that a(n) > 0 for all n = 2..5*10^7.
%C A301452 It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)}| - |{d>0: d|n and d == 5,7 (mod 8)}|.
%H A301452 Zhi-Wei Sun, <a href="/A301452/b301452.txt">Table of n, a(n) for n = 1..10000</a>
%H A301452 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H A301452 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e A301452 a(2) = 2 since 2^2 = 2*4^0 + 0^2 + 2*1^2 and 2^2 = 3*4^0 + 1^2 + 2*0^2.
%e A301452 a(3) = 2 since 3^2 = 2*4^1 + 1^2 + 2*0^2 and 3^2 = 3*4^0 + 2^2 + 2*1^2.
%e A301452 a(5) = 2 since 5^2 = 2*4^1 + 3^2 + 2*2^2 and 5^2 = 3*4^0 + 2^2 + 2*3^2.
%t A301452 f[n_]:=f[n]=n/2^(IntegerExponent[n,2]);
%t A301452 OD[n_]:=OD[n]=Divisors[f[n]];
%t A301452 QQ[n_]:=QQ[n]=(n==0)||(n>0&&Sum[JacobiSymbol[-2,Part[OD[n],i]],{i,1,Length[OD[n]]}]!=0);
%t A301452 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A301452 tab={};Do[r=0;Do[If[QQ[n^2-m*4^k],Do[If[SQ[n^2-m*4^k-2x^2],r=r+1],{x,0,Sqrt[(n^2-m*4^k)/2]}]],{m,2,3},{k,0,Log[4,n^2/m]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A301452 Cf. A000290, A000302, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391.
%K A301452 nonn
%O A301452 1,2
%A A301452 _Zhi-Wei Sun_, Mar 21 2018