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%I #22 Jul 21 2023 06:54:25
%S 2,2,2,1,2,3,2,2,2,1,3,4,3,2,3,5,3,1,4,3,3,4,3,2,2,5,5,1,3,2,4,3,3,3,
%T 2,6,3,2,1,3,6,4,2,2,3,5,3,3,1,2,4,3,4,2,5,4,5,5,3,2,7,4,4,3,2,6,3,4,
%U 4,3,9,3,2,3,3,6,5,4,4,3,8,5,2,2,3,6,3,3,4,4,5,7,2,3,3,8,6,1,5,4
%N Number of ways to write 12*n+1 as x^2 + 4*y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer.
%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4, 10, 18, 28, 39, 49, 98, 142, 163, 184, 208, 320, 382, 408, 814, 910, 1414, 2139, 2674, 3188, 3213, 4230, 6279, 25482.
%C (ii) All the numbers 16*n+5 (n = 0,1,2,...) can be written as x^4 + 4*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.
%C (iii) All the numbers 24*n+1 (n = 0,1,2,...) can be written as 12*x^4 + 4*y^2 + z^2 with x,y,z integers. Also, all the numbers 24*n+9 (n = 0,1,2,...) can be written as 2*x^4 + 6*y^2 + z^2 with x,y,z positive integers.
%C (iv) All the numbers 24*n+2 (n = 0,1,2,...) can be written as x^4 + 9*y^2 + z^2, where x,y,z are integers with z > 0. Also, all the numbers 24*n+17 (n = 0,1,2,...) can be written as x^4 + 16*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.
%C (v) All the numbers 30*n+3 (n = 1,2,3,...) can be written as 2*x^4 + 3*y^2 + z^2 with x,y,z positive integers. Also, all the numbers 30*n+21 (n = 0,1,2,...) can be written as 2*x^4 + 3*y^2 + z^2, where x,y,z are integers with z > 0.
%H Zhi-Wei Sun, <a href="/A290491/b290491.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Sci. China Math. 58 (2015), 1367-1396.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2</a>, arXiv:1502.03056 [math.NT], 2015-2017.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), No. 2, 97-120.
%e a(10) = 1 since 12*10+1 = 7^2 + 4*4^2 + 8*1^4.
%e a(28) = 1 since 12*28+1 = 9^2 + 4*8^2 + 8*0^4.
%e a(49) = 1 since 12*49+1 = 19^2 + 4*5^2 + 8*2^4.
%e a(3188) = 1 since 12*3188+1 = 103^2 + 4*80^2 + 8*4^4.
%e a(3213) = 1 since 12*3213+1 = 91^2 + 4*87^2 + 8*0^4.
%e a(4230) = 1 since 12*4230+1 = 223^2 + 4*16^2 + 8*1^4.
%e a(6279) = 1 since 12*6279+1 = 19^2 + 4*75^2 + 8*9^4.
%e a(25482) = 1 since 12*25482+1 = 531^2 + 4*58^2 + 8*6^4.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[12n+1-8x^4-4y^2],r=r+1],{x,0,((12n+1)/8)^(1/4)},{y,1,Sqrt[(12n+1-8x^4)/4]}];Print[n," ",r],{n,1,100}]
%Y Cf. A000290, A000583, A270566, A286885, A286944, A287616, A290342, A290472.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 03 2017
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