A253941 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, (p^6 + 5)/6, (p^8 + 5)/6 and (p^10 + 5)/6 are all prime.

184279409, 619338131, 913749803, 1057351301, 1507289869, 1600204213, 2845213937, 4725908767, 4760956439, 5374709801, 5518707641, 8724256757, 9044067313, 9387396269, 10992352517, 11937043567, 13493126359, 13593105793, 17891702891, 17897035213, 17954907767, 19690938161, 20227580927, 20922685813, 21313027583, 21717176851

Offset: 1

Author

Zak Seidov, Jan 20 2015

Keywords

• nonn

Comments

The sequence contains all terms up to 10^10. There are no terms as yet for which (p^12 + 5)/6 is also prime.

No terms < 10^11 with (p^12 + 5)/6 prime. - Chai Wah Wu, Jan 27 2015

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..67

Crossrefs

Subsequence of A253976.

Cf. A118915, A247478, A253925, A253940.

Programs

PARI

lista(nn) = forprime(p=5, nn, if(ispseudoprime((p^2 + 5)/6) && ispseudoprime((p^4 + 5)/6) && ispseudoprime((p^6 + 5)/6) && ispseudoprime((p^8 + 5)/6) && ispseudoprime((p^10 + 5)/6), print1(p, ", "))); \\ Jinyuan Wang, Mar 01 2020

Python

from gmpy2 import is_prime, t_divmod
A253941_list = []
for p in range(1,10**6,2):
    if is_prime(p):
        p2, x = p**2, 1
        for i in range(5):
            x *= p2
            q, r = t_divmod(x+5,6)
            if r or not is_prime(q):
                break
        else:
            A253941_list.append(p) # Chai Wah Wu, Jan 22 2015

Extensions

a(15)-a(26) from Chai Wah Wu, Jan 22 2015