A238597 Number of primes p < 2*n with 2*pi(p) + 1 and p*(2n-1) - 2 both prime, where pi(.) is given by A000720.

0, 1, 2, 1, 1, 2, 2, 4, 1, 1, 5, 3, 1, 3, 1, 2, 4, 3, 3, 2, 2, 3, 4, 3, 1, 5, 3, 1, 3, 2, 4, 5, 2, 2, 2, 3, 3, 6, 3, 3, 4, 2, 4, 5, 3, 4, 5, 3, 2, 6, 2, 3, 8, 1, 1, 5, 5, 3, 5, 4, 4, 6, 2, 3, 3, 4, 3, 7, 3, 1, 7, 4, 4, 5, 4, 3, 8, 4, 1, 7

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 for no n > 195.

(ii) For any integer n > 1, there is a prime p < 2*n with 2*pi(p) + 1 (or 2*pi(p) - 1) and 2*n + p both prime.

Part (i) of this conjecture is an extension of the conjecture in A238580.

Examples

			a(9) = 1 since 5, 2*pi(5) + 1 = 2*3 + 1 = 7 and 5*(2*9-1) - 2 = 5*17 - 2 = 83 are all prime.
a(28) = 1 since 3, 2*pi(3) + 1 = 2*2 + 1 = 5 and 3*(2*28-1) - 2 = 3*55 - 2 = 163 are all prime.
a(195) = 1 since 71, 2*pi(71) + 1 = 2*20 + 1 = 41 and 71*(2*195-1) - 2 = 27617 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, A000720, A237284, A238580.

Programs

Mathematica

p[n_,k_]:=p[n,k]=PrimeQ[k]&&PrimeQ[2*PrimePi[k]+1]&&PrimeQ[k*(2n-1)-2]
a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,2n-1}]
Table[a[n],{n,1,80}]