A065897 The a(n)-th composite number is twice the n-th prime.

1, 2, 5, 7, 13, 16, 22, 25, 31, 41, 43, 52, 59, 62, 69, 78, 87, 91, 101, 107, 111, 120, 127, 137, 149, 155, 159, 166, 170, 177, 199, 206, 215, 218, 235, 239, 248, 259, 266, 277, 286, 289, 306, 309, 316, 319, 339, 359, 366, 369, 375, 386, 389, 406, 416, 426, 438

Offset: 1

Author

Labos Elemer, Nov 28 2001

Keywords

• nonn
• easy

Comments

Also the least k such that the n-th primorial (A002110) is a divisor of the k-th compositorial (A036691). - Reinhard Zumkeller, Sep 03 2002

Examples

			a(7) = 22 because twice the 7th prime (2*17 = 34) is the 22nd composite number: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000720, A002110, A002808, A036691, A100484 (even semiprimes).

Programs

Magma

A065897:= func< n | 2*NthPrime(n) -1 -#PrimesUpTo(2*NthPrime(n)) >;
[A065897(n): n in [1..130]]; // G. C. Greubel, Aug 24 2024

Maple

A065897:=n->2*ithprime(n)-(numtheory[pi](2*ithprime(n)))-1: seq(A065897(n), n=1..100); # Wesley Ivan Hurt, Sep 16 2017

Mathematica

Table[2*Prime[n]-(PrimePi[2*Prime[n]])-1, {n, 128}]

PARI

{ for (n=1, 1000, f=2*prime(n); a=f - primepi(f) - 1; write("b065897.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 04 2009

SageMath

def A065897(n): return 2*nth_prime(n) -prime_pi(2*nth_prime(n)) -1
[A065897(n) for n in range(1,131)] # G. C. Greubel, Aug 24 2024

Formula

a(n) = 2*prime(n) - (pi(2*prime(n))) - 1, where pi = A000720.