A303656 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.

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, 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, 7, 4, 5, 5, 5, 4, 6, 5, 6, 10, 5, 4, 5, 7, 4, 9, 2, 9, 8, 5, 6, 6

Offset: 1

Author

Zhi-Wei Sun, Apr 27 2018

Keywords

• nonn

Comments

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.

It has been verified that a(n) > 0 for all n = 2..2*10^10.

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.

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

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

Examples

			a(2) = 1 with 2 = 0^2 + 0^2 + 3^0 + 5^0.
a(5) = 1 with 5 = 0^2 + 1^2 + 3^1 + 5^0.
a(25) = 1 with 25 = 1^2 + 4^2 + 3^1 + 5^1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
• 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. 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.

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[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]