A396480 Primes p such that p-1 has primitive roots (i.e., is in A033948).

2, 3, 5, 7, 11, 19, 23, 47, 59, 83, 107, 163, 167, 179, 227, 251, 263, 347, 359, 383, 467, 479, 487, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1459, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2663

Offset

1,1

Comments

2, 3, 5, along with primes p such that (p-1)/2 is an odd prime power.

Any rational prime p coprime to N decomposes into a product of EulerPhi(N)/ord(p,N) prime ideals in Z[zeta_N], where ord(p,N) is the multiplicative order of p modulo N. As a result, this sequence lists primes P such that: for every Dirichlet character chi modulo P, there exist some rational primes p which remain prime in Z[chi], where Z[chi] is the ring generated by values of chi.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Wikipedia, Dirichlet character.

Example

The Dirichlet characters modulo 19 have orders 1, 2, 3, 6, 9, or 18. In both Z[zeta_3] = Z[zeta_6] and Z[zeta_9] = Z[zeta_18] there are rational primes remaining inert (namely primes p == 2 (mod 3) in the former and p == 2, 5 (mod 9) in the latter), so 19 is a term.

Prog

(PARI) isoddprimepower(n) = (n%2) && isprimepower(n)
isA396480(p) = isprime(p) && (p<=5 || isoddprimepower((p-1)/2))

Crossrefs

Cf. A002322, A033948, A061345 (powers of odd primes), A396478, A396481 (complement in primes).

Sequence in context: A059878 A105017 A214197 * A083771 A158069 A383781

Adjacent sequences: A396477 A396478 A396479 * A396481 A396482 A396483

Keyword

nonn,easy

Author

Jianing Song, May 27 2026

Status

approved