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A300448 Number of the integers 4^k*m with k >= 0 and m = 1, 2, 5 such that n^2 - 4^k*m can be written as the sum of two squares.

%I A300448 #19 Mar 06 2018 12:53:19
%S A300448 1,2,4,2,4,5,4,2,7,5,6,5,6,5,5,2,5,7,6,5,10,7,5,5,7,7,6,5,6,6,7,2,10,
%T A300448 6,8,7,7,6,6,5,7,11,5,7,10,5,6,5,8,8,9,7,8,6,6,5,10,7,5,6,5,8,7,2,5,
%U A300448 10,8,6,10,8,7,7,11,7,7,6,10,7,5,5
%N A300448 Number of the integers 4^k*m with k >= 0 and m = 1, 2, 5 such that n^2 - 4^k*m can be written as the sum of two squares.
%C A300448 Conjecture 1: a(n) > 0 for all n > 0.
%C A300448 Conjecture 2: Each positive square n^2 can be written as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5 17.
%C A300448 Conjecture 3: For each n = 1,2,3,... we can write n^2 as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5, 65.
%C A300448 Conjecture 3 implies that A300441(n) > 0 for all n > 0 since  4*A001353(0)^2 + 1 = 1, 4*A001353(1)^2 + 1 = 5 and 4*A001353(2)^2 + 1 = 65.
%C A300448 We have verified Conjectures 1-3 for all n = 1..2*10^7.
%H A300448 Zhi-Wei Sun, <a href="/A300448/b300448.txt">Table of n, a(n) for n = 1..10000</a>
%H A300448 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 A300448 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 A300448 a(1) = 1 since 1^2 - 4^0*1 = 0 = 0^2 + 0^2.
%e A300448 a(2) = 2 since 2^2 - 4^0*2 = 2 = 1^2 + 1^2 and 2^2 - 4*1 = 0 = 0^2 + 0^2.
%e A300448 a(4) = 2 since 4^2 - 4^1*2 = 8 = 2^2 + 2^2 and 4^2 - 4^2*1 = 0 = 0^2 + 0^2.
%t A300448 f[n_]:=f[n]=FactorInteger[n];
%t A300448 g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1,1,0],{i,1,Length[f[n]]}]==0;
%t A300448 QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
%t A300448 u[1]=1; u[2]=2; u[3]=5;
%t A300448 tab={};Do[r=0;Do[If[QQ[n^2-4^k*u[m]],r=r+1],{m,1,3},{k,0,Log[4,n^2/u[m]]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A300448 Cf. A000118, A000302, A001481, A271518, A281976, A299924, A299537, A299794, A300362, A300441, A300396.
%K A300448 nonn
%O A300448 1,2
%A A300448 _Zhi-Wei Sun_, Mar 06 2018