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

1,3

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000720, A237284, A238580.

Sequence in context: A309224 A103784 A153916 * A045870 A309890 A229037

Adjacent sequences: A238594 A238595 A238596 * A238598 A238599 A238600

Keyword

nonn

Author

Zhi-Wei Sun, Mar 01 2014

Status

approved