A282426 Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 4, 2, 1, 5, 6, 5, 4, 5, 4, 5, 3, 4, 7, 5, 5, 7, 5, 3, 5, 6, 5, 6, 4, 5, 5, 6, 4, 7, 7, 6, 9, 6, 5, 6, 5, 4, 6, 4, 4, 7, 6, 6, 6, 5, 2, 6, 3, 4, 6, 6, 5, 6, 6, 6, 10, 7, 5

Offset: 1

Author

Zhi-Wei Sun, Feb 14 2017

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) Each n = 1,2,3,... can be written as x^4 + y^2 + z^2 + 2^k with x,y,z nonnegative integers and k among 0,1,2,3,4, 5.

(iii) Any positive integer n can be written as 4*x^4 + y^3 + z^2 + 3^k with k,x,y,z nonnegative integers.

We have verified parts (i) and (ii)-(iii) for n up to 2*10^7 and 10^7 respectively.

See arXiv:1701.05868 for more such conjectures.

Examples

			a(1) = 1 since 1 = 0^4 + 4*0^2 + 0^2 + 3^0.
a(16) = 1 since 16 = 0^4 + 4*1^2 + 3^2 + 3^1.
a(475) = 1 since 475 = 3^4 + 4*6^2 + 13^2 + 3^4.
a(556) = 1 since 556 = 0^4 + 4*0^2 + 23^2 + 3^3.
a(8641) = 1 since 8641 = 9^4 + 4*21^2 + 17^2 + 3^3.
a(52696) = 1 since 52696 = 12^4 + 4*87^2 + 41^2 + 3^1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Crossrefs

Cf. A000079, A000244, A000290, A000583, A280356.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-4y^2-3^k],r=r+1],{k,0,Min[4,Log[3,n]]},{x,0,(n-3^k)^(1/4)},{y,0,Sqrt[(n-3^k-x^4)/4]}];Print[n," ",r];Continue,{n,1,80}]