A291635 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2*y + 5*z, p - 2, p + 4 and p + 10 are all prime.

0, 1, 1, 0, 1, 3, 1, 0, 1, 2, 2, 0, 3, 7, 3, 0, 4, 4, 1, 0, 4, 7, 3, 0, 3, 5, 2, 0, 4, 6, 2, 0, 2, 3, 3, 0, 4, 8, 3, 0, 5, 8, 2, 0, 2, 5, 2, 0, 5, 8, 4, 0, 4, 5, 2, 0, 5, 6, 4, 0, 1, 8, 5, 0, 3, 9, 3, 0, 6, 8, 3, 0, 5, 13, 5, 0, 9, 9, 2, 0, 4, 6, 6, 0, 7, 11, 4, 0, 8, 10, 5, 0, 2, 11, 5, 0, 3, 10, 4, 0

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 1 not divisible by 4.

This is stronger than the conjecture in A291624. Obviously, it implies that there are infinitely many prime quadruples (p-2, p, p+4, p+10).

We have verified that a(n) > 0 for any integer 1 < n < 10^7 not divisible by 4.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..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(61) = 1 since 61 = 4^2 + 0^2 + 3^2 + 6^2 with 4 + 2*0 + 5*3 = 19, 19 - 2 = 17, 19 + 4 = 23 and 19 + 10 = 29 all prime.
a(253) = 1 since 253 = 12^2 + 8^2 + 3^2 + 6^2 with 12 + 2*8 + 5*3 = 43, 43 - 2 = 41, 43 + 4 = 47 and 43 + 10 = 53 all prime.
a(725) = 1 since 725 = 7^2 + 0^2 + 0^2 + 26^2 with 7 + 2*0 + 5*0 = 7, 7 - 2 = 5, 7 + 4 = 11 and 7 + 10 = 17 all prime.
a(1511) = 1 since 1511 = 18^2 + 15^2 + 11^2 + 29^2 with 18 + 2*15 + 5*11 = 103, 103 - 2 = 101, 103 + 4 = 107 and 103 + 10 = 113 all prime.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[p_]:=PQ[p]=PrimeQ[p]&&PrimeQ[p-2]&&PrimeQ[p+4]&&PrimeQ[p+10]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&PQ[x+2y+5z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}]; Print[n, " ", r], {n, 1, 100}]

Crossrefs

Cf. A000040, A000118, A000290, A022004, A172454, A271518, A281976, A290935, A291150, A291191, A291455, A291624.

Sequence in context: A031253 A378929 A291624 * A308243 A268386 A122779

Adjacent sequences: A291632 A291633 A291634 * A291636 A291637 A291638

Keyword

nonn,look

Author

Zhi-Wei Sun, Aug 28 2017

Status

approved