cp's OEIS Frontend

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

%I A281980 #9 Feb 04 2017 11:23:36
%S A281980 0,1,2,5,16,26,32,36,50,80,81,90,145,162,226,256,260,356,405,416,485,
%T A281980 512,576,625,626,641,661,677,746,800,821,981,1066,1226,1250,1280,1296,
%U A281980 1440,1601,1781,2020,2106,2146,2320,2401,2410,2426,2501,2570,2592,2602,2801,2916,2977,3125,3250,3490,3616,3761,3845
%N A281980 Numbers of the form x^4 + y^2 with x^2 + 24*y a square, where x and y are nonnegative integers.
%C A281980 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.
%C A281980 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.
%H A281980 Zhi-Wei Sun, <a href="/A281980/b281980.txt">Table of n, a(n) for n = 1..5000</a>
%H A281980 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%e A281980 a(1) = 0 since 0 = 0^4 + 0^2 with 0^2 + 24*0 = 0^2.
%e A281980 a(2) = 1 since 1 = 1^4 + 0^2 with 1^2 + 24*0 = 1^2.
%e A281980 a(3) = 2 since 2 = 1^4 + 1^2 with 1^2 + 24*1 = 5^2.
%e A281980 a(4) = 5 since 5 = 1^4 + 2^2 with 1^2 + 24*2 = 7^2.
%e A281980 a(5) = 16 since 16 = 2^4 + 0^2 with 2^2 + 24*0 = 2^2.
%e A281980 a(6) = 26 since 26 = 1^4 + 5^2 with 1^2 + 24*5 = 11^2.
%t A281980 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A281980 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}]
%Y A281980 Cf. A000290, A000583, A281976.
%K A281980 nonn
%O A281980 1,3
%A A281980 _Zhi-Wei Sun_, Feb 04 2017