A291123 Number of ways to write 4*n+1 as (p-1)^2 + q^2 + r^2, where p is prime and q and r are nonnegative integers with q <= r.

1, 2, 2, 1, 3, 3, 1, 3, 3, 2, 5, 3, 2, 4, 2, 3, 4, 4, 2, 3, 4, 2, 4, 1, 2, 8, 4, 2, 3, 5, 3, 5, 5, 2, 4, 4, 4, 6, 5, 3, 8, 4, 3, 6, 1, 5, 7, 6, 2, 5, 6, 3, 8, 3, 3, 5, 5, 5, 4, 4, 5, 6, 3, 2, 7, 8, 4, 8, 5, 2, 8, 5, 3, 8, 7, 4, 6, 4, 3, 4, 8

Offset

0,2

Comments

Conjecture: (i) a(n) > 0 for all n, and a(n) = 1 only for n = 0, 3, 6, 23, 44.

(ii) Any positive integer relatively prime to 6 can be written as (p-1)^2 + q^2 + 3*r^2, where p is prime, and q and r are integers.

(iii) Let n be any nonnegative integer. Then 4*n+1 can be written as x^2 + y^2 + z^2, where x,y,z are nonnegative integers with x + y + 2*z prime; 4*n+2 can be written as x^2 + y^2 + z^2, where x,y,z are nonnegative integers with x + 3*y prime. Also, we can write 4*n+3 as 2*x^2 + y^2 + z^2, where x,y,z are nonnegative integers with 2*x+1 prime.

(iv) Let n be any nonnegative integer. Then we can write 4*n+1 as x^2 + y^2 + z^2 with x,y,z integers such that x^2 + 5*y^2 + 7*z^2 prime, and write 4*n+2 as x^2 + y^2 + z^2 with x,y,z integers such that x^2 + 2*y^2 + 5*z^2 prime. Also, we can write 8*n+3 as x^2 + y^2 + z^2 with x,y,z integers such that 2*x^2 + 4*y^2 + 5*z^2 is prime.

Note that by the Gauss-Legendre theorem any positive integer not of the form 4^k*(8*m+7) (k,m = 0,1,2,...) can be written as the sum of three squares.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Example

a(0) = 1 since 4*0+1 = (2-1)^2 + 0^2 + 0^2 with 2 prime.
a(3) = 1 since 4*3+1 = (3-1)^2 + 0^2 + 3^2 with 3 prime.
a(6) = 1 since 4*6+1 = (5-1)^2 + 0^2 + 3^2 with 5 prime.
a(23) = 1 since 4*23+1 = (3-1)^2 + 5^2 + 8^2 with 3 prime.
a(44) = 1 since 4*44+1 = (3-1)^2 + 2^2 + 13^2 with 3 prime.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[PrimeQ[p]&&SQ[4n+1-(p-1)^2-q^2], r=r+1], {p, 2, Sqrt[4n+1]+1}, {q, 0, Sqrt[(4n+1-(p-1)^2)/2]}]; Print[n, " ", r], {n, 0, 80}]

Crossrefs

Cf. A000040, A000290, A260418, A271518, A290472, A290491, A291047.

Sequence in context: A220073 A272020 A264263 * A292594 A093613 A344086

Adjacent sequences: A291120 A291121 A291122 * A291124 A291125 A291126

Keyword

nonn

Author

Zhi-Wei Sun, Aug 17 2017

Status

approved