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

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 09 2019

STATUS

approved