A023247 Primes p such that 3*p + 4 and 9*p + 16 are also prime.

3, 5, 23, 59, 73, 79, 109, 179, 269, 373, 383, 389, 409, 439, 509, 599, 683, 709, 929, 983, 1019, 1129, 1193, 1409, 1423, 1453, 1663, 1699, 1879, 2039, 2053, 2069, 2579, 2753, 2963, 3049, 3169, 3203, 3259, 3719, 3769, 3833, 4799, 4973, 4993, 5303, 5443, 5483

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Original name: Numbers n such that n remains prime through 2 iterations of the function f(x) = 3x + 4.

n, 3*n + 4, 9*n + 16 are primes. - Vincenzo Librandi, Aug 04 2010

Except for a(2) = 5, all terms are congruent to 3 or 9 (mod 10). If p == 1 (mod 10), 3p + 4 == 7 (mod 10) could be prime, but then 9p + 16 == 5 (mod 10). - Alonso del Arte, Nov 23 2018

Examples

			3 * 3 + 4 = 11, which is prime, and 3 * 11 + 4 = 37, which is also prime, so 3 is in the sequence.
3 * 5 + 4 = 19, which is prime, and 3 * 19 + 4 = 61, which is also prime, so 5 is in the sequence.
3 * 7 + 4 = 25 = 5^2, so 7 is not in the sequence.
Although 3 * 11 + 4 = 37, which is prime, 3 * 37 = 115 = 5 * 23, so 11 is not in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040 (primes), A016777 (3n+1, so A016777(n+1) = 3n+4).

Programs

Magma

[n: n in [0..100000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16)]; // Vincenzo Librandi, Aug 04 2010

Mathematica

Select[Prime[Range[800]], And@@PrimeQ[Rest[NestList[3# + 4 &, #, 2]]] &] (* Harvey P. Dale, Jan 21 2014 *)

PARI

select(is(p)=isprime(3*p+4)&&isprime(9*p+16)&&isprime(p), primes([2,5500])) \\ Defines the is() function. The select() command provides a check & illustration. isprime(p) at the end improves performance if a selection is operated on primes as here. - M. F. Hasler, Nov 23 2018

Extensions

Better name from M. F. Hasler, Nov 23 2018