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%I #30 Jul 22 2022 01:26:49
%S 0,0,0,0,0,1,2,2,2,2,4,3,4,5,5,5,4,4,5,4,5,9,8,6,6,9,7,6,8,8,10,8,4,8,
%T 5,7,9,12,9,6,10,9,11,10,8,16,12,8,9,12,9,11,12,11,9,10,12,14,10,10
%N Number of ways to write n as x^2 + y^2 + 2^z + 5*2^w, where x,y,z,w are nonnegative integers with x <= y.
%C Conjecture: a(n) > 0 for all n > 5.
%C This has been verified for n up to 5*10^9. Note that 321256731 cannot be written as x^2 + (2*y)^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.
%C In contrast, Crocker 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.
%C 570143 cannot be written as x^2 + y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers, and 2284095 cannot be written as x^2 + y^2 + 2^z + 7*2^w with x,y,z,w nonnegative integers.
%C Jiao-Min Lin (a student at Nanjing University) found a counterexample to the conjecture: a(18836421387) = 0. - _Zhi-Wei Sun_, Jul 21 2022
%D R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
%H Zhi-Wei Sun, <a href="/A303637/b303637.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://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(6) = 1 with 6 = 0^2 + 0^2 + 2^0 + 5*2^0.
%e a(8) = 2 with 8 = 1^2 + 1^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^1 + 5*2^0.
%e a(9) = 2 with 9 = 1^2 + 1^2 + 2^1 + 5*2^0 = 0^2 + 0^2 + 2^2 + 5*2^0.
%e a(10) = 2 with 10 = 0^2 + 2^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^2 + 5*2^0.
%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[n-5*2^k-2^m],Do[If[SQ[n-5*2^k-2^m-x^2],r=r+1],{x,0,Sqrt[(n-5*2^k-2^m)/2]}]],{k,0,Log[2,n/5]},{m,0,If[n/5==2^k,-1,Log[2,n-5*2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]
%Y Cf. A000079, A000290, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303639, A303656.
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Apr 27 2018
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