A384187 Primes p such that p + 6, p^2 + 6, p^3 + 6, p^4 + 6 and p^5 + 6 are primes.

1361, 70216961, 71317991, 311153281, 371383381, 385230821, 400675721, 466490881, 487757861, 620258761, 818694271, 822486341, 888942491, 898259491, 1102784471, 1423261241, 1443957371, 1623698051, 1628827091, 1729743571, 1831375171, 1837091231, 1904579381, 1978478521, 2070333781

Offset

1,1

Comments

This is a subsequence of A023201: primes p such that p + 6 is also prime (sexy primes).

The largest tuple of primes of the form (p, p + m, p^2 + m, p^3 + m,..., p^k + m), where m is a digit from 1 to 9, is the 6-tuple (p, p + 6, p^2 + 6, p^3 + 6, p^4 + 6, p^5 + 6). Indeed, if m is an odd digit, then p must be 2 and the longest tuple is (2, 2+3, 2^2+3, 2^3+3, 2^4+3).

For p > 2 we consider the cases:

If m = 2, it is satisfied that p^2 + 2 == 0 (mod 3) for all p!= 3. In fact, (p, p + 2, p^2 + 2) are prime only if p = 3.

If m = 4, then p^4 + 4 == 0 (mod 5), for all p!= 5. Then the longest tuple is (p, p + 4, p^2 + 4, p^3 + 4), which are the p primes of A243734

If m = 6, then p^6 + 6 == 0 (mod 7), for all p!= 7. Thus, the largest tuple is (p, p + 6, p^2 + 6, p^3 + 6, p^4 + 6, p^5 + 6), where a(n) gives these primes.

If m = 8, then p^2 + 8 == 0 (mod 3) for all p!=3. In fact, (p, p + 8, p^2 + 8) are prime only if p = 3.

Links

Table of n, a(n) for n=1..25.

Mathematica

Select[Prime[Range[32327000]], AllTrue[#^Range[0, 5]+6, PrimeQ]&] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Jun 13 2025 *)

Prog

(Python)
from sympy import isprime, primerange
lim = 10**9
A384187 = []
for p in primerange(2, lim):
    if isprime(p + 6) and isprime(p**2 + 6) and isprime(p**3 + 6) and isprime(p**4 + 6) and isprime(p**5 + 6):
       A384187.append(p)
print(", ".join(str(p) for p in A384187))

Crossrefs

Cf. A023201, A243734.

Sequence in context: A242458 A223254 A260441 * A265498 A113507 A119521

Adjacent sequences: A384184 A384185 A384186 * A384188 A384189 A384190

Keyword

nonn

Author

Gonzalo Martínez, May 21 2025

Status

approved