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%I #17 Apr 23 2018 04:04:28
%S 0,1,2,2,3,3,2,4,6,3,5,6,4,6,7,4,4,9,6,6,8,4,9,9,5,7,7,5,7,9,4,8,13,7,
%T 6,11,7,10,13,8,9,10,7,9,11,7,9,15,8,8,14,6,9,16,6,8,11,11,10,12,8,7,
%U 15,10,8,11,9,14,15,9
%N Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.
%C Conjecture: a(n) > 0 for all n > 1.
%C This is stronger than the author's conjecture in A303233. I have verified a(n) > 0 for all n = 2..10^9.
%C In contrast, Corcker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.
%H Zhi-Wei Sun, <a href="/A303363/b303363.txt">Table of n, a(n) for n = 1..10000</a>
%H R. C. Crocker, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/112/2/87281/on-the-sum-of-two-squares-and-two-powers-of-k">On the sum of two squares and two powers of k</a>, Colloq. Math. 112(2008), 235-267.
%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://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 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 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
%e a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&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 tab={};Do[r=0;Do[If[QQ[4(n-4^j-2^k)+1],Do[If[SQ[8(n-4^j-2^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-4^j-2^k)+1]-1)/2}]],{j,0,Log[4,n/2]},{k,2j,Log[2,n-4^j]}];tab=Append[tab,r],{n,1,70}];Print[tab]
%Y Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303389.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 22 2018
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