A349661 Number of ways to write n as x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.

1, 3, 3, 2, 4, 5, 2, 2, 4, 5, 6, 4, 3, 6, 4, 1, 6, 7, 4, 6, 8, 5, 1, 4, 6, 10, 10, 3, 6, 10, 2, 3, 8, 6, 10, 10, 5, 6, 4, 5, 12, 14, 6, 5, 9, 6, 2, 3, 6, 12, 14, 7, 5, 8, 2, 7, 14, 6, 9, 9, 5, 9, 4, 2, 10, 15, 7, 7, 8, 7, 3, 5, 5, 7, 14, 5, 9, 9, 1, 4, 11, 8, 11, 13, 7, 13, 7, 2, 11, 17, 12, 8, 5, 6, 7, 5, 7, 11, 12, 8

Offset

1,2

Comments

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

This has been verified for n up to 10^6.

As (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2, the conjecture gives a new refinement of Lagrange's four-square theorem.

See also A350012 for a similar conjecture.

Links

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

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167--190.

Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893.  See also arXiv:1701.05868 [math.NT].

Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Example

a(1) = 1 with 1 = 0^4 + 0^2 + (1^2 + 4^0)/2.
a(23) = 1 with 23 = 1^4 + 3^2 + (5^2 + 4^0)/2.
a(79) = 1 with 79 = 1^4 + 2^2 + (12^2 + 4^1)/2.
a(1199) = 1 with 1199 = 5^4 + 18^2 + (22^2 + 4^2)/2.
a(3679) = 1 with 3679 = 5^4 + 2^2 + (78^2 + 4^2)/2.
a(6079) = 1 with 6079 = 3^4 + 42^2 + (92^2 + 4^1)/2.
a(33439) = 1 with 33439 = 1^4 + 175^2 + (75^2 + 4^0)/2.
a(50399) = 1 with 50399 = 13^4 + 135^2 + (85^2 + 4^0)/2.
a(207439) = 1 with 207439 = 1^4 + 142^2 + (612^2 + 4^1)/2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[2(n-x^4-y^2)-4^z], r=r+1], {x, 0, (n-1)^(1/4)}, {y, 0, Sqrt[n-1-x^4]}, {z, 0, Log[4, 2(n-x^4-y^2)]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000118, A000290, A000302, A000583, A349957, A349992, A350012.

Sequence in context: A106702 A303338 A271510 * A282545 A306471 A354698

Adjacent sequences: A349658 A349659 A349660 * A349662 A349663 A349664

Keyword

nonn

Author

Zhi-Wei Sun, Dec 08 2021

Status

approved