A308547 Number of ways to write n as a^2 + 2*b^2 + 2^c*3^d, where a,b,c,d are nonnegative integers.

1, 2, 3, 4, 4, 4, 4, 4, 5, 7, 5, 8, 6, 5, 6, 4, 6, 8, 7, 10, 8, 6, 4, 8, 8, 8, 10, 10, 6, 9, 6, 4, 10, 9, 11, 14, 8, 8, 9, 10, 8, 11, 8, 9, 13, 6, 5, 8, 9, 10, 11, 13, 7, 14, 8, 10, 13, 9, 11, 16, 7, 7, 13, 4, 12, 12, 10, 12, 10, 13, 5, 14, 13, 9, 17, 12, 7, 12, 6, 10

Offset: 1

Author

Zhi-Wei Sun, Jun 06 2019

Keywords

• nonn

Comments

As 3*(a^2 + 2*b^2 + 2^c*3^d) = (a+2*b)^2 + 2*(a-b)^2 + 2^c*3^(d+1), we have a(3*n) > 0 if a(n) > 0.

The first positive integer n with a(n) = 0 is 139571911. We also have a(142991573) = 0.

Examples

			a(1) = 1 with 1 = 0^2 + 2*0^2 + 2^0*3^0.
a(2) = 2 with 2 = 0^2 + 2*0^2 + 2^1*3^0 = 1^2 + 2*0^2 + 2^0*3^0.
a(1117) = 2 with 1117 = 10^2 + 2*12^2 + 2^0*3^6 = 19^2 + 2*18^2 + 2^2*3^3.
a(78373) = 1 with 78373 = 271^2 + 2*48^2 + 2^2*3^4.
a(448159) = 1 with 448159 = 610^2 + 2*195^2 + 2^0*3^2.
a(82816213) = 2 with 82816213 = 4353^2 + 2*5651^2 + 2^1*3^0 = 3681^2 + 2*5885^2 + 2^1*3^0.
a(90685253) = 2 with 90685253 = 7007^2 + 2*4560^2 + 2^2*3^0 = 607^2 + 2*6720^2 + 2^2*3^0.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000244, A000290, A002479, A302984, A303637, A303656, A308411.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-3^k*2^m-2x^2],r=r+1],{k,0,Log[3,n]},{m,0,Log[2,n/3^k]},{x,0,Sqrt[(n-3^k*2^m)/2]}];tab=Append[tab,r],{n,1,80}];Print[tab]