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%I #42 Jul 30 2022 12:45:39
%S 0,1,1,2,1,3,2,3,2,4,3,4,2,4,4,3,2,4,4,3,2,4,3,4,1,4,5,6,4,6,5,5,6,6,
%T 5,8,4,6,6,5,4,7,5,7,5,6,4,5,3,4,7,6,7,8,5,4,7,5,5,9,3,6,5,6,4,6,5,7,
%U 7,4,5,5,5,4,6,5,6,10,5,4,5,7,4,9,2,9,8,5,6,6
%N Number of ways to write n as a^2 + b^2 + 3^c + 5^d, where a,b,c,d are nonnegative integers with a <= b.
%C Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares, a power of 3 and a power of 5.
%C It has been verified that a(n) > 0 for all n = 2..2*10^10.
%C It seems that any integer n > 1 also can be written as the sum of two squares, a power of 2 and a power of 3.
%C The author would like to offer 3500 US dollars as the prize for the first proof of his conjecture that a(n) > 0 for all n > 1. - _Zhi-Wei Sun_, Jun 05 2018
%C Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 2.4*10^11. - _Zhi-Wei Sun_, Jul 30 2022
%H Zhi-Wei Sun, <a href="/A303656/b303656.txt">Table of n, a(n) for n = 1..100000</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(2) = 1 with 2 = 0^2 + 0^2 + 3^0 + 5^0.
%e a(5) = 1 with 5 = 0^2 + 1^2 + 3^1 + 5^0.
%e a(25) = 1 with 25 = 1^2 + 4^2 + 3^1 + 5^1.
%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[n-3^k-5^m],Do[If[SQ[n-3^k-5^m-x^2],r=r+1],{x,0,Sqrt[(n-3^k-5^m)/2]}]],{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, A303429, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303702, A303821.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Apr 27 2018
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