A070046 Number of primes between prime(n) and 2*prime(n) exclusive.

1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 9, 9, 9, 11, 13, 12, 13, 14, 13, 15, 15, 16, 19, 20, 19, 19, 18, 18, 23, 23, 25, 25, 27, 26, 28, 28, 28, 28, 30, 30, 32, 32, 32, 32, 35, 38, 38, 38, 39, 39, 39, 41, 42, 43, 42, 42, 42, 42, 42, 44, 49, 50, 49, 49, 54, 54, 56, 55, 55, 55, 57, 58

Offset: 1

Author

Enoch Haga, May 05 2002

Keywords

• easy
• nonn

Examples

			a(1)=1 because between p=2 and 4 there is exactly one prime, 3.
a(10)=6 since six consecutive primes (31,37,41,43,47,53) are located between p(10) = 29 and 58.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Bertrand's Postulate

Crossrefs

Cf. A060715, A077463, A246514.

Related sequences:

Primes (p) and composites (c): A000040, A002808, A000720, A065855.

Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.

Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

Programs

Maple

N:= 1000: # to get a(n) for n <= pi(N)
Primes:=select(isprime,[$1..N]):
seq(numtheory:-pi(2*Primes[n])-n, n=1..nops(Primes)); # Robert Israel, Aug 28 2014

Mathematica

pp[n_]:=Module[{pr=Prime[n]},PrimePi[2pr]-n]; Array[pp,80] (* Harvey P. Dale, Mar 30 2015; edited by Zak Seidov, Oct 18 2022  *)

PARI

forprime(p=2, 5000, n=0; for(q=p+1, 2*p-1, if(isprime(q), n++)); print1(n, ", ")) \\ Harry J. Smith, Dec 13 2007, improved by Colin Barker, Aug 28 2014

PARI

a(n)=primepi(2*prime(n))-n \\ Charles R Greathouse IV, Aug 28 2014

Python

from sympy import prime, primepi
def A070046(n): return primepi(prime(n)<<1)-n # Chai Wah Wu, Oct 22 2024

Formula

a(n) = primepi(2*prime(n))-n. - Charles R Greathouse IV, Aug 28 2014

a(n) = A060715(A000040(n)).

a(n) = A063124(n)-1. - N. J. A. Sloane, Oct 19 2024

Extensions

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar