A303363 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

0, 1, 2, 2, 3, 3, 2, 4, 6, 3, 5, 6, 4, 6, 7, 4, 4, 9, 6, 6, 8, 4, 9, 9, 5, 7, 7, 5, 7, 9, 4, 8, 13, 7, 6, 11, 7, 10, 13, 8, 9, 10, 7, 9, 11, 7, 9, 15, 8, 8, 14, 6, 9, 16, 6, 8, 11, 11, 10, 12, 8, 7, 15, 10, 8, 11, 9, 14, 15, 9

Offset: 1

Author

Zhi-Wei Sun, Apr 22 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

This is stronger than the author's conjecture in A303233. I have verified a(n) > 0 for all n = 2..10^9.

In contrast, Corcker 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.

Examples

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Crossrefs

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303389.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
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;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-4^j-2^k)+1],Do[If[SQ[8(n-4^j-2^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-4^j-2^k)+1]-1)/2}]],{j,0,Log[4,n/2]},{k,2j,Log[2,n-4^j]}];tab=Append[tab,r],{n,1,70}];Print[tab]