A281980 Numbers of the form x^4 + y^2 with x^2 + 24*y a square, where x and y are nonnegative integers.

0, 1, 2, 5, 16, 26, 32, 36, 50, 80, 81, 90, 145, 162, 226, 256, 260, 356, 405, 416, 485, 512, 576, 625, 626, 641, 661, 677, 746, 800, 821, 981, 1066, 1226, 1250, 1280, 1296, 1440, 1601, 1781, 2020, 2106, 2146, 2320, 2401, 2410, 2426, 2501, 2570, 2592, 2602, 2801, 2916, 2977, 3125, 3250, 3490, 3616, 3761, 3845

Offset: 1

Author

Zhi-Wei Sun, Feb 04 2017

Keywords

• nonn

Comments

If m and x are integers with m == x or -x (mod 4) and m == x or -x (mod 3), then y = (m^2-x^2)/24 is an integer with x^2 + 24*y = m^2. So, the sequence has infinitely many terms.

The conjecture in A281976 implies that any nonnegative integer can be written as the sum of two squares and a term of the current sequence.

Examples

			a(1) = 0 since 0 = 0^4 + 0^2 with 0^2 + 24*0 = 0^2.
a(2) = 1 since 1 = 1^4 + 0^2 with 1^2 + 24*0 = 1^2.
a(3) = 2 since 2 = 1^4 + 1^2 with 1^2 + 24*1 = 5^2.
a(4) = 5 since 5 = 1^4 + 2^2 with 1^2 + 24*2 = 7^2.
a(5) = 16 since 16 = 2^4 + 0^2 with 2^2 + 24*0 = 2^2.
a(6) = 26 since 26 = 1^4 + 5^2 with 1^2 + 24*5 = 11^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Crossrefs

Cf. A000290, A000583, A281976.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
n=0;Do[Do[If[SQ[m-x^4]&&SQ[x^2+24*Sqrt[m-x^4]],n=n+1;Print[n," ",m];Goto[aa]],{x,0,m^(1/4)}];Label[aa];Continue,{m,0,4000}]