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

1,2

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A072358 A074795 A069577 * A318781 A375834 A357186

Adjacent sequences: A282423 A282424 A282425 * A282427 A282428 A282429

Keyword

nonn

Author

Zhi-Wei Sun, Feb 14 2017

Status

approved