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A302984 Number of ways to write n as x^2 + 2*y^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.

%I A302984 #13 Apr 19 2018 12:35:36
%S A302984 0,0,0,0,0,1,2,2,3,3,3,4,5,5,8,5,5,7,4,6,7,9,9,10,10,7,9,8,10,15,10,9,
%T A302984 10,8,6,10,10,11,14,14,8,12,13,13,20,15,12,16,10,15,12,10,15,17,16,12,
%U A302984 16,14,14,21
%N A302984 Number of ways to write n as x^2 + 2*y^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.
%C A302984 Conjecture: a(n) > 0 for all n > 5.
%C A302984 Clearly, a(2*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 6...10^9.
%C A302984 See also A302982 and A302983 for similar conjectures.
%H A302984 Zhi-Wei Sun, <a href="/A302984/b302984.txt">Table of n, a(n) for n = 1..10000</a>
%H A302984 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 A302984 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%H A302984 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 A302984 a(6) = 1 with 6 = 0^2 + 2*0^2 + 2^0 + 5*2^0.
%e A302984 a(7) = 2 with 7 = 1^2 + 2*0^2 + 2^0 + 5*2^0 = 0^2 + 2*0^2 + 2^1 + 5*2^0.
%e A302984 a(8) = 2 with 8 = 0^2 + 2*1^2 + 2^0 + 5*2^0 = 1^2 + 2*0^2 + 2^1 + 5*2^0.
%t A302984 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A302984 f[n_]:=f[n]=FactorInteger[n];
%t A302984 g[n_]:=g[n]=Sum[Boole[MemberQ[{5,7},Mod[Part[Part[f[n],i],1],8]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t A302984 QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t A302984 tab={};Do[r=0;Do[If[QQ[n-5*2^k-2^j],Do[If[SQ[n-5*2^k-2^j-2x^2],r=r+1],{x,0,Sqrt[(n-5*2^k-2^j)/2]}]],{k,0,Log[2,n/5]},{j,0,Log[2,Max[1,n-5*2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]
%Y A302984 Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302985.
%K A302984 nonn
%O A302984 1,7
%A A302984 _Zhi-Wei Sun_, Apr 16 2018