A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 10, 10, 10, 10, 12, 14, 13, 14, 15, 14, 16, 16, 17, 20, 21, 20, 20, 19, 19, 24, 24, 26, 26, 28, 27, 29, 29, 29, 29, 31, 31, 33, 33, 33, 33, 36, 39, 39, 39, 40, 40, 40, 42, 43, 44, 43, 43, 43, 43, 43, 45, 50, 51, 50, 50, 55, 55, 57, 56, 56, 56, 58

Offset: 1

Author

Reinhard Zumkeller, Aug 08 2001

Keywords

• nonn

Comments

a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - Sean A. Irvine, Apr 18 2023

Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - Antti Karttunen, Dec 08 2024

Examples

			a(10) = 7 as there are 7 primes between prime(10) = 29 and 58 = 29*2: 29, 31, 37, 41, 43, 47, 53.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 2000 terms from Harry J. Smith]

Crossrefs

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.

Cf. A003961, A108227, A331677, A334051, A378746.

Programs

Maple

A062134 := proc(n) numtheory:-pi(2*ithprime(n))-n+1; end; # N. J. A. Sloane, Oct 19 2024
[seq(A062134(n),n=1..100)];

Mathematica

Table[PrimePi[2*Prime[n]] - n + 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

PARI

a(n)={1 + primepi(2*prime(n)) - n} \\ Harry J. Smith, Aug 19 2009

Formula

a(n) = A035250(prime(n)).

a(n) = A070046(n) + 1. - Sean A. Irvine, Apr 18 2023

From Antti Karttunen, Dec 08 2024: (Start)

a(n) = n-A331677(n) = 1+n-A334051(n).

a(n) = 1+A000720(2*A000040(n))-n. [After Harry J. Smith's PARI-program]

a(n) < A108227(n). [Assuming M. F. Hasler's interpretation in May 08 2017 comment in the latter]

a(n) = A001222(A378746(n)).

(End)

Extensions

Definition clarified by N. J. A. Sloane, Oct 04 2024