A238643 Number of primes p <= n such that 2*pi(p) - (-1)^n and p*n +((-1)^n - 3)/2 are both prime, where pi(x) is the number of primes not exceeding x.

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

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 2, then 2*p*n + 1 (or 2*p*n - 1) is prime for some prime p < n.

Part (i) of the conjecture is a further extension of the conjecture in A238597 to cover the even case.

Examples

			a(9) = 1 since 5, 2*pi(5)-(-1)^9 = 2*3 + 1 = 7 and 5*9 + ((-1)^9-3)/2 = 45 - 2 = 43 are all prime.
a(10) = 1 since 3, 2*pi(3)-(-1)^(10) = 2*2 - 1 = 3 and 3*10 + ((-1)^(10)-3)/2 = 30 - 1 = 29 are all prime.
a(268) = 1 since 23, 2*pi(23) - 1`= 2*9 - 1 = 17 and 23*268 - 1 = 6163 are all prime.
a(389) = 1 since 71, 2*pi(71) + 1 = 2*20 + 1 = 41 and 71*389 - 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, A238580, A238597.

Programs

Mathematica

p[n_,k_]:=PrimeQ[2k-(-1)^n]&&PrimeQ[n*Prime[k]+((-1)^n-3)/2]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n]}]
Table[a[n],{n,1,80}]