A308028 Number of ways to write 2*n+1 as p + q + r with 2*p + 4*q + 6*r a square, where p,q,r are odd primes.

0, 0, 0, 1, 0, 0, 2, 1, 1, 3, 3, 3, 1, 2, 2, 5, 2, 4, 6, 4, 3, 3, 6, 7, 4, 4, 5, 2, 5, 7, 5, 8, 3, 7, 7, 6, 6, 10, 6, 12, 8, 7, 8, 12, 7, 9, 14, 9, 6, 8, 10, 7, 10, 13, 9, 12, 11, 12, 16, 12, 12, 13, 10, 13, 14, 13, 12, 14, 13, 13, 16, 12, 13, 20, 16, 11, 12, 13, 12, 19, 18, 12, 17, 21, 12, 19, 17, 11, 19, 17, 18, 18, 20, 12, 23, 17, 13, 18, 18, 14

Offset

1,7

Comments

2-4-6 Conjecture: a(n) > 0 for all n > 6. In other words, any odd integer greater than 14 can be written as the sum of three odd primes p,q,r for which 2*p + 4*q + 6*r is an integer square.

This is stronger than the solved weak Goldbach conjecture (A068307), and it is motivated by the author's 1-3-5 conjecture in A271518.

We have verified a(n) > 0 for all n = 7..3*10^5.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

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

Zhi-Wei Sun, Write 2n+1 > 14 as p+q+r with p,q,r odd primes and 2p+4q+6r a square, Question 331170 on MathOverflow, May 10, 2019.

Example

a(8) = 1 with 2*8+1 = 17 = 7 + 5 + 5 and 2*7 + 4*5 + 6*5 = 8^2.
a(9) = 1 with 2*9+1 = 19 = 11 + 3 + 5 and 2*11 + 4*3 + 6*5 = 8^2.
a(13) = 1 with 2*13+1 = 7 + 17 + 3 and 2*7 + 4*17 + 6*3 = 10^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; p[n_]:=p[n]=Prime[n];
tab={}; Do[r=0; Do[If[PrimeQ[2n+1-p[i]-p[j]]&&SQ[2p[i]+4p[j]+6(2n+1-p[i]-p[j])], r=r+1], {i, 2, PrimePi[2n]}, {j, 2, PrimePi[2n-p[i]]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000040, A068307, A271518.

Sequence in context: A284834 A279677 A262180 * A356077 A320902 A189913

Adjacent sequences: A308025 A308026 A308027 * A308029 A308030 A308031

Keyword

nonn

Author

Zhi-Wei Sun, May 09 2019

Status

approved