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

1,2

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.

Links

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

Example

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.

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]

Crossrefs

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

Sequence in context: A287422 A120510 A029117 * A195850 A253593 A174901

Adjacent sequences: A308544 A308545 A308546 * A308548 A308549 A308550

Keyword

nonn

Author

Zhi-Wei Sun, Jun 06 2019

Status

approved