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%I #16 Jul 30 2022 11:56:36
%S 0,0,0,1,1,1,3,3,2,4,3,2,6,2,4,8,2,4,7,3,4,8,5,5,10,6,4,10,8,5,12,7,3,
%T 12,4,5,12,5,5,14,7,4,12,7,6,12,6,6,10,7,7,12,7,6,14,6,8,16,4,8,18,5,
%U 6,16,5,9,13,7,7,14
%N Number of ways to write n as x^2 + 2*y^2 + 3*2^z + 4^w with x,y,z,w nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 3.
%C This is stronger than the author's previous conjecture in A302983. It has been verified that a(n) > 0 for all n = 4..10^9.
%C Jiao-Min Lin (a student at Nanjing University) has found a counterexample to the conjecture: a(12558941213) = 0. - _Zhi-Wei Sun_, Jul 30 2022
%H Zhi-Wei Sun, <a href="/A303338/b303338.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(4) = 1 with 4 = 0^2 + 2*0^2 + 3*2^0 + 4^0.
%e a(5) = 1 with 5 = 1^2 + 2*0^2 + 3*2^0 + 4^0.
%e a(6) = 1 with 6 = 0^2 + 2*1^2 + 3*2^0 + 4^0.
%e a(9) = 2 with 9 = 0^2 + 2*1^2 + 3*2^0 + 4^1 = 0^2 + 2*1^2 + 3*2^1 + 4^0.
%t f[n_]:=f[n]=FactorInteger[n];
%t 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 QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[QQ[n-3*2^k-4^j],Do[If[SQ[n-3*2^k-4^j-2x^2],r=r+1],{x,0,Sqrt[(n-3*2^k-4^j)/2]}]],{k,0,Log[2,n/3]},{j,0,If[3*2^k==n,-1,Log[4,n-3*2^k]]}];tab=Append[tab,r],{n,1,70}];Print[tab]
%Y Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303363.
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Apr 22 2018
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