A301391 - OEIS

%I #13 Mar 20 2018 16:11:18

%S 1,1,2,1,1,2,1,1,4,1,1,2,1,1,2,1,3,4,1,1,8,2,2,2,3,2,6,1,2,2,1,1,11,3,

%T 2,4,4,3,3,1,6,10,6,2,7,2,3,2,6,3,8,2,7,7,2,1,11,4,4,2,2,1,6,1,3,11,3,

%U 3,16,3,5,4,8,5,2,3,11,5,8,1

%N Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y even such that x^2 - (6*y)^2 = 4^k for some k = 0,1,2,....

%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 11, 13, 19, 2^k*m (k = 0,1,2,... and m = 1, 5, 7, 31).

%C We have verified a(n) > 0 for all n = 1..10^7.

%C See also A301376 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A301391/b301391.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 a(2) = 1 since 2^2 = 2^2 + 0^2 + 0^2 + 0^2 with 2^2 - (6*0)^2 = 4^1.

%e a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4^2 - (6*0)^2 = 4^2.

%e a(7) = 1 since 7^2 = 2^2 + 0^2 + 3^2 + 6^2 with 2^2 - (6*0)^2 = 4^1.

%e a(11) = 1 since 11 = 2^2 + 0^2 + 6^2 + 9^2 with 2^2 - (6*0)^2 = 4^1.

%e a(13) = 1 since 13 = 4^2 + 0^2 + 3^2 + 12^2 with 4^2 - (6*0)^2 = 4^2.

%e a(19) = 1 since 19 = 1^2 + 0^2 + 6^2 + 18^2 with 1^2 - (6*0)^2 = 4^0.

%e a(31) = 1 since 31^2 = 20^2 + 2^2 + 14^2 + 19^2 with 20^2 - (6*2)^2 = 4^4.

%e a(75) = 2 since 75^2 = 68^2 + 10^2 + 1^2 + 30^2 = 68^2 + 10^2 + 15^2 + 26^2 with 68^2 - (6*10)^2 = 4^5.

%t f[n_]:=f[n]=FactorInteger[n];

%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;

%t QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[4^k+144m^2]&&QQ[n^2-4^k-148m^2], Do[If[SQ[n^2-(4^k+148m^2)-z^2],r=r+1],{z,0,Sqrt[(n^2-4^k-148m^2)/2]}]],{k,0,Log[2,n]},{m,0,Sqrt[(n^2-4^k)/148]}];tab=Append[tab,r],{n,1,80}];Print[tab]

%Y Cf. A000118, A000290, A000302, A299537, A299794, A299924, A300219, A300396, A300441, A300510, A301376.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 20 2018