A350012 Number of ways to write n as 4*x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.

1, 2, 1, 1, 4, 4, 1, 3, 5, 5, 3, 3, 4, 7, 3, 2, 6, 5, 2, 4, 6, 2, 2, 5, 4, 6, 2, 2, 6, 7, 2, 2, 6, 5, 5, 4, 3, 7, 5, 5, 8, 6, 2, 6, 9, 4, 2, 4, 5, 8, 3, 3, 5, 8, 3, 6, 5, 3, 6, 4, 6, 5, 6, 1, 10, 9, 2, 6, 11, 8, 1, 7, 5, 11, 6, 4, 7, 10, 3, 6, 10, 4, 8, 8, 6, 8, 6, 5, 11, 13, 5, 1, 11, 8, 3, 4, 4, 9, 7, 6

Offset

1,2

Comments

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

This is a new refinement of Lagrange's four-square theorem since (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2. We have verified the conjecture for n up to 10^6.

See also A349661 for a similar conjecture.

We also have some other conjectures of such a type.

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) = 4*0^4 + 0^2 + (1^2 + 4^0)/2.
a(3) = 1 with 3 = 4*0^4 + 1^2 + (0^2 + 4)/2.
a(4) = 1 with 4 = 4*0^4 + 0^2 + (2^2 + 4)/2.
a(7) = 1 with 7 = 4*1^4 + 1^2 + (0^2 + 4)/2.
a(71) = 1 with 71 = 4*1^4 + 3^2 + (10^2 + 4^2)/2.
a(92) = 1 with 92 = 4*1^4 + 6^2 + (10^2 + 4)/2.
a(167) = 1 with 167 = 4*1^4 + 9^2 + (10^2 + 4^3)/2.
a(271) = 1 with 271 = 4*1^4 + 11^2 + (6^2 + 4^4)/2.
a(316) = 1 with 316 = 4*1^4 + 4^2 + (24^2 + 4^2)/2.
a(4796) = 1 with 4796 = 4*5^4 + 36^2 + (44^2 + 4^3)/2.
a(14716) = 1 with 14716 = 4*5^4 + 4^2 + (156^2 + 4^3)/2.
a(24316) = 1 with 24316 = 4*3^4 + 84^2 + (184^2 + 4^2)/2.

Mathematica

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

Crossrefs

Cf. A000118, A000290, A000302, A000583, A349661, A349957, A349992.

Sequence in context: A260625 A306614 A264336 * A322038 A123521 A322115

Adjacent sequences: A350009 A350010 A350011 * A350013 A350014 A350015

Keyword

nonn

Author

Zhi-Wei Sun, Dec 08 2021

Status

approved