

A023247


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


4



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
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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



