cp's OEIS Frontend

A338162 Number of ways to write 4*n + 1 as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some k = 0,1,2,..., where x, y, z, w are nonnegative integers with z <= w.

%I A338162 #15 Oct 15 2020 03:40:48
%S A338162 1,2,3,2,4,4,2,3,6,3,7,6,5,6,7,3,8,9,5,6,8,4,8,7,4,8,11,3,7,8,6,8,13,
%T A338162 6,6,8,6,9,11,8,10,12,7,7,12,5,14,14,7,7,13,5,13,13,5,8,13,8,10,10,7,
%U A338162 13,10,6,9,14,9,10,15,7,10
%N A338162 Number of ways to write 4*n + 1 as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some k = 0,1,2,..., where x, y, z, w are nonnegative integers with z <= w.
%C A338162 Conjecture: a(n) > 0 for all n >= 0. Moreover, if m > 1 has the form 2^a*(2*b+1), and either a is positive and even, or b is even,  then m can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some positive integer k, where x, y, z, w are nonnegative integers.
%C A338162 We have verified the latter assertion in the conjecture for m up to 4*10^8.
%H A338162 Zhi-Wei Sun, <a href="/A338162/b338162.txt">Table of n, a(n) for n = 0..10000</a>
%H A338162 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. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.
%H A338162 Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893.  See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.
%H A338162 Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.
%e A338162 a(0) = 1, and 4*0 + 1 = 1^2 + 0^2 + 0^2 +0^2 with 1^2 + 7*0^2 = 2^0.
%e A338162 a(25) = 2, and 25 = 2^2 + 2^2 + 1^2 + 4^2 = 4^2 + 0^2 + 0^2 + 3^2
%e A338162 with 2^2 + 7*2^2 = 2^5 and 4^2 + 7*0^2 = 2^4.
%t A338162 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A338162 PQ[n_]:=PQ[n]=IntegerQ[Log[2,n]];
%t A338162 tab={};Do[r=0;Do[If[SQ[4n+1-x^2-y^2-z^2]&&PQ[x^2+7y^2],r=r+1],{x,1,Sqrt[4n+1]},{y,0,Sqrt[4n+1-x^2]},{z,0,Sqrt[(4n+1-x^2-y^2)/2]}];tab=Append[tab,r],{n,0,70}];tab
%Y A338162 Cf. A000079, A000118, A000290, A020670, A337082, A338094, A338095, A338096, A338139.
%K A338162 nonn
%O A338162 0,2
%A A338162 _Zhi-Wei Sun_, Oct 14 2020