A282226 Least nonnegative integer m which can be written in exactly n ways as x^2 + y^2 + z^2 + w^2 with both x and x + 24*y squares, where x,y,z,w are nonnegative integers with z <= w.

0, 1, 2, 10, 18, 34, 41, 52, 66, 100, 90, 130, 261, 306, 226, 370, 426, 405, 612, 585, 661, 626, 770, 666, 756, 706, 810, 981, 882, 1026, 1266, 1170, 1330, 1530, 1476, 1426, 1881, 1701, 2650, 2410, 2506, 1666, 1386, 2226, 3861, 2626, 3366, 3006, 2106, 2610, 3346, 3186, 3226, 4410, 3786, 3850, 2826, 3762, 4026, 4500

Offset: 1

Author

Zhi-Wei Sun, Feb 09 2017

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0.

See also A281976 for a related conjecture.

Examples

			a(1) = 0 since 0 = 0^2 + 0^2 + 0^2 + 0^2 with 0 = 0^2 and 0 + 24*0 = 0^2.
a(2) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0^2 and 0 + 24*0 = 0^2, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 = 1^2 and 1 + 24*0 = 1^2.
a(4) = 10 since 10 = 0^2 + 0^2 + 1^2 + 3^2 with 0 = 0^2 and 0 + 24*0 = 0^2, 10 = 1^2 + 0^2 + 0^2 + 3^2 with 1 = 1^2 and 1 + 24*0 = 1^2, 10 = 1^2 + 1^2 + 2^2 + 2^2 with 1 = 1^2 and 1 + 24*1 = 5^2, and 10 = 1^2 + 2^2 + 1^2 + 2^2 with 1 = 1^2 and 1 + 24*2 = 7^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..300
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Crossrefs

Cf. A000118, A000290, A270969, A281976, A281980.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[m=0;Label[aa];r=0;Do[If[SQ[m-x^4-y^2-z^2]&&SQ[x^2+24y],r=r+1;If[r>n,m=m+1;Goto[aa]]],{x,0,m^(1/4)},{y,0,Sqrt[m-x^4]},{z,0,Sqrt[(m-x^4-y^2)/2]}];If[r