A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 6, 5, 5, 5, 3, 4, 6, 6, 7, 6, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 6, 6, 4, 5, 5, 5, 7, 6, 6, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 3, 4, 5, 5

Offset: 1

Author

Zhi-Wei Sun, Mar 03 2014

Keywords

• nonn

Comments

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.

In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.

Examples

			a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A045917, A046927, A049002, A062326, A230502, A237413, A237705.

Programs

Mathematica

PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
p[n_,k_]:=PQ[Floor[(n-Prime[k])/4]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]