A049233 Primes p such that p + 2 is squarefree.

3, 5, 11, 13, 17, 19, 29, 31, 37, 41, 53, 59, 67, 71, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 163, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 337, 347, 353, 379, 383, 389

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

A001359 (lesser of twin primes) is a subsequence. - Michel Marcus, Aug 10 2018

This sequence is infinite and its relative density in the sequence of primes is equal to 2 * Product_{p prime} (1-1/(p*(p-1))) = 2 * A005596 = 0.747911... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Denis Xavier Charles, Sieve Methods, Master's Thesis, 2000, p. 93.
• Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Crossrefs

Cf. A001359, A005117, A005596.

Programs

Maple

A049233:=n->`if`(isprime(n) and numtheory[issqrfree](n+2), n, NULL): seq(A049233(n), n=1..600); # Wesley Ivan Hurt, Nov 22 2015

Mathematica

Select[Prime[Range[100]], SquareFreeQ[#+2]&] (* Jean-François Alcover, Nov 22 2015 *)

PARI

isok(p) = isprime(p) && issquarefree(p+2); \\ Michel Marcus, Dec 31 2013

Extensions

Definition simplified by Michel Marcus, Dec 31 2013