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

1,3

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000290, A000583, A281976.

Sequence in context: A117557 A175735 A275172 * A137997 A213359 A328000

Adjacent sequences: A281977 A281978 A281979 * A281981 A281982 A281983

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2017

Status

approved