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A281980 Numbers of the form x^4 + y^2 with x^2 + 24*y a square, where x and y are nonnegative integers. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A323006

Adjacent sequences:  A281977 A281978 A281979 * A281981 A281982 A281983

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 04 2017

STATUS

approved

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Last modified October 14 04:37 EDT 2019. Contains 327995 sequences. (Running on oeis4.)