A303432 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.

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, 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, 8, 8, 5, 7, 4, 6, 7, 9, 8, 9, 7, 8

Offset: 1

Author

Zhi-Wei Sun, Apr 23 2018

Keywords

• nonn

Comments

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.

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.

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.

All the three conjectures hold for n = 2..2*10^6. Note that either of them is stronger than the conjecture in A303233.

See also A303363, A303389 and A303401 for similar conjectures.

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
• 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.

Crossrefs

Cf. A000079, A000384, A014105, A303233, A303338, A303363, A303389, A303393, A303399, A303401.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
HexQ[n_]:=HexQ[n]=SQ[8n+1]&&(n==0||Mod[Sqrt[8n+1]+1,4]==0);
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-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]