A396481 Primes p such that p-1 does not have primitive roots (i.e., is in A033949).

13, 17, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401

Offset

1,1

Comments

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: there exist some Dirichlet characters chi modulo P such that no rational primes p remain prime in Z[chi], where Z[chi] is the ring generated by values of chi.

Links

Table of n, a(n) for n=1..59.

Wikipedia, Dirichlet character.

Example

13 is a term since there are Dirichlet characters modulo 13 of order 12, and no rational primes remain inert in Z[zeta_12].
17 is a term since there are Dirichlet characters modulo 17 of order 8 or 16, and in either Z[zeta_8] or Z[zeta_16] there are no rational primes remaining inert.

Prog

(PARI) isoddprimepower(n) = (n%2) && isprimepower(n)
isA396481(p) = isprime(p) && (p>5 && !isoddprimepower((p-1)/2))

Crossrefs

Cf. A002322, A033948, A061345 (powers of odd primes), A396479, A396480 (complement in primes).

Sequence in context: A283217 A283390 A078138 * A164074 A152426 A152427

Adjacent sequences: A396478 A396479 A396480 * A396482 A396483 A396484

Keyword

nonn,easy

Author

Jianing Song, May 27 2026

Status

approved