A290491 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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Aug 03 2017

Keywords

• nonn

Comments

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.

(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.

(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.

(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.

(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.

Examples

			a(10) = 1 since 12*10+1 = 7^2 + 4*4^2 + 8*1^4.
a(28) = 1 since 12*28+1 = 9^2 + 4*8^2 + 8*0^4.
a(49) = 1 since 12*49+1 = 19^2 + 4*5^2 + 8*2^4.
a(3188) = 1 since 12*3188+1 = 103^2 + 4*80^2 + 8*4^4.
a(3213) = 1 since 12*3213+1 = 91^2 + 4*87^2 + 8*0^4.
a(4230) = 1 since 12*4230+1 = 223^2 + 4*16^2 + 8*1^4.
a(6279) = 1 since 12*6279+1 = 19^2 + 4*75^2 + 8*9^4.
a(25482) = 1 since 12*25482+1 = 531^2 + 4*58^2 + 8*6^4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), 1367-1396.
• Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), No. 2, 97-120.

Crossrefs

Cf. A000290, A000583, A270566, A286885, A286944, A287616, A290342, A290472.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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}]