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%I #9 Apr 24 2018 05:51:51
%S 0,1,2,3,3,3,2,3,4,5,4,4,2,3,3,4,5,7,5,5,4,4,4,7,5,4,3,2,2,4,5,7,8,7,
%T 5,7,5,7,7,7,4,4,2,3,5,7,6,9,7,6,5,6,5,7,7,3,3,3,3,5,7,7,8,7,6,8,5,8,
%U 8,8,5,7,4,6,7,9,8,9,7,8
%N Number of ways to write n as a*(2*a-1) + b*(2*b-1) + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.
%C Conjecture 1: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two hexagonal numbers and two powers of 2.
%C Conjecture 2: Any integer n > 1 can be written as a*(2*a+1) + b*(2*b+1) + 2^c + 2^d with a,b,c,d nonnegative integers.
%C Conjecture 3: Each integer n > 1 can be written as a*(2*a-1) + b*(2*b+1) + 2^c + 2^d with a,b,c,d nonnegative integers.
%C All the three conjectures hold for n = 2..2*10^6. Note that either of them is stronger than the conjecture in A303233.
%C See also A303363, A303389 and A303401 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A303432/b303432.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://math.scichina.com:8081/sciAe/EN/abstract/abstract517007.shtml">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), no. 7, 1367-1396.
%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.
%e a(2) = 1 with 2 = 0*(2*0-1) + 0*(2*0-1) + 2^0 + 2^0.
%e a(7) = 2 with 7 = 1*(2*1-1) + 1*(2*1-1) + 2^0 + 2^2 = 0*(2*0-1) + 1*(2*1-1) + 2^1 + 2^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t HexQ[n_]:=HexQ[n]=SQ[8n+1]&&(n==0||Mod[Sqrt[8n+1]+1,4]==0);
%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-2^j-2^k)+1],Do[If[HexQ[n-2^j-2^k-x(2x-1)],r=r+1],{x,0,(Sqrt[4(n-2^j-2^k)+1]+1)/4}]],{j,0,Log[2,n/2]},{k,j,Log[2,n-2^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000079, A000384, A014105, A303233, A303338, A303363, A303389, A303393, A303399, A303401.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 23 2018
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