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

1,1

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

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A075307 A100302 A290384 * A027753 A066411 A153410

Adjacent sequences: A023244 A023245 A023246 * A023248 A023249 A023250

Keyword

nonn

Author

David W. Wilson

Extensions

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

Status

approved