A396479 - OEIS

%I #7 May 28 2026 19:24:01

%S 13,17,25,26,29,31,32,34,35,37,39,41,43,45,49,50,51,52,53,55,58,61,62,

%T 64,65,67,68,70,71,73,74,75,77,78,79,82,85,86,87,89,90,91,93,95,96,97,

%U 98,99,100,101,102,103,104,105,106,109,110,111,112,113,115,116,117,119

%N Numbers k such that psi(k) does not have primitive roots (i.e., is in A033949), where psi = A002322.

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_character">Dirichlet character</a>.

%e 13 is a term since there are Dirichlet characters modulo 13 of order 12, and no rational primes remain inert in Z[zeta_12].

%e 32 is a term since there are Dirichlet characters modulo 32 of order 8, and no rational primes remain inert in Z[zeta_8].

%o (PARI) isoddprimepower(n) = (n%2) && isprimepower(n)

%o isA396479(n) = (240%n!=0) && !isoddprimepower(znstar(n)[2][1]/2)

%Y Cf. A002322, A033949, A061345 (powers of odd primes), A396478 (complement), A396481 (prime terms).

%K nonn,easy

%O 1,1

%A _Jianing Song_, May 27 2026