A274525 Prime numbers p such that p - 2, p^2 - p - 1, p^2 - p + 1 are prime numbers.

7, 139, 1789, 2731, 4159, 5641, 13339, 13399, 19429, 21739, 22369, 32059, 32911, 33601, 42571, 45319, 54541, 55339, 65449, 68821, 106189, 108499, 111871, 132859, 136399, 138079, 141511, 142981, 148201, 149629, 152041, 152839, 173431, 174049, 178249

Offset

1,1

Links

Pierre CAMI, Table of n, a(n) for n = 1..30000

Example

5 - 2 = 3 prime, 5 prime, 5*(5-1) - 1 = 19 prime, 5*(5-1) + 1 = 21 composite, so 5 is not in the sequence.
7 - 2 = 5 prime, 7 prime, 7*(7-1) - 1 = 41 prime, 7*(7-1) + 1 = 43 prime so 7 is in the sequence.

Mathematica

Select[Prime[Range[100]], Union[PrimeQ[{# - 2, #^2 - # - 1, #^2 - # + 1}]] == {True} &] (* Alonso del Arte, Jun 27 2016 *)
Select[Prime[Range[17000]], AllTrue[{#-2, #^2-#-1, #^2-#+1}, PrimeQ]&] (* Harvey P. Dale, Jun 20 2024 *)

Prog

(PFGW & SCRIPT) twin.txt file with the smallest of twin pairs.
SCRIPT
DIM n
OPENFILEIN maf, twin.txt
OPENFILEOUT myf, a.txt
LABEL loop
GETNEXT n, maf
SET n, n+2
SET m, n*(n-1)-1
PRP m
IF ISPRP THEN GOTO a
GOTO loop
LABEL a
PRP m+2
IF ISPRP THEN WRITE myf, n
GOTO loop
(PARI) lista(nn) = forprime(p=2, nn, if (isprime(p-2) && isprime(p^2-p-1) && isprime(p^2-p+1), print1(p, ", "))); \\ Michel Marcus, Jul 07 2016
(Python)
from sympy import isprime, primerange
def aupto(n):
    t = []
    for p in primerange(2, n+1):
        if isprime(p-2) and isprime(p**2 - p - 1) and isprime(p**2 - p + 1):
            t.append(p)
    return t # Paul Muljadi, Jun 21 2024

Crossrefs

Cf. A228968.

Sequence in context: A142295 A056254 A137463 * A221375 A351334 A190195

Adjacent sequences: A274522 A274523 A274524 * A274526 A274527 A274528

Keyword

nonn

Author

Pierre CAMI, Jun 27 2016

Status

approved