A275675 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+5*z^2, where k,x,y,z are nonnegative integers with x <= y.

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 3, 2, 2, 3, 2, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 2, 3, 3, 1, 3, 3, 2, 1, 2, 1, 5, 3, 2, 2, 3, 4, 1, 4, 2, 3, 5, 2, 2, 3, 1, 3, 3, 1, 4, 2, 4, 1, 2, 3, 2, 6, 2, 3, 3, 2, 2, 2, 2, 2, 3

Offset

1,6

Comments

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

This is stronger than the known fact that any natural number can be written as w^2 + x^2 + y^2 + 5*z^2 with w,x,y,z integers.

See also A275656, A275676 and A275678 for similar conjectures.

Links

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

Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Example

a(43) = 1 since 43 = 4^0*(1+1^2+6^2) + 5*1^2.
a(45) = 1 since 45 = 4*(1+0^2+3^2) + 5*1^2.
a(237) = 1 since 237 = 4^3*(1+1^2+1^2) + 5*3^2.
a(561) = 1 since 561 = 4*(1+8^2+8^2) + 5*3^2.
a(9777) = 1 since 9777 = 4*(1+11^2+31^2) + 5*33^2.
a(39108) = 1 since 39108 = 4^2*(1+11^2+31^2) + 5*66^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[(n-4^k*(1+x^2+y^2))/5], r=r+1], {k, 0, Log[4, n]}, {x, 0, Sqrt[(n/4^k-1)/2]}, {y, x, Sqrt[n/4^k-1-x^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000118, A000290, A271518, A275648, A275656, A275676, A275678.

Sequence in context: A120256 A300647 A114811 * A219025 A270746 A043531

Adjacent sequences: A275672 A275673 A275674 * A275676 A275677 A275678

Keyword

nonn

Author

Zhi-Wei Sun, Aug 04 2016

Status

approved