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%I #16 Apr 28 2018 11:31:25
%S 0,1,1,2,1,3,2,3,2,4,3,4,2,4,4,3,2,4,4,3,2,4,3,4,1,4,4,6,3,6,4,5,5,6,
%T 4,8,4,6,5,5,4,7,5,7,5,6,4,5,3,4,6,5,5,7,5,3,6,4,4,8,3,6,5,5,4,6,4,7,
%U 6,4,4,5,4,4,5,4,5,8,4,4,5,6,4,8,2,9,7,5,5,6
%N Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares.
%C Conjecture: a(n) > 0 for all n > 1.
%C This is equivalent to the author's conjecture in A303656.
%C It has been verified that a(n) > 0 for all n = 2..10^9.
%H Zhi-Wei Sun, <a href="/A303429/b303429.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.
%p a(5) = 1 with 5 - 3^1 - 5^0 = 0^2 + 1^2.
%p a(25) = 1 with 25 - 3^1 - 5^1 = 1^2 + 4^2.
%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-3^k-5^m],r=r+1],{k,0,Log[3,n]},{m,0,If[n==3^k,-1,Log[5,n-3^k]]}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Apr 28 2018
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