OFFSET
1,1
COMMENTS
Also norms of prime ideals in Z[sqrt(6)], which is a unique factorization domain.
Consists of 2, 3, the primes p == 1, 5, 19, 23 (mod 24) and the squares of primes p such that p == 7, 11, 13, 17 (mod 24).
There are two distinct prime ideals with norm p == 1, 5, 19, 23 (mod 24), which decomposes in Q(sqrt(3)). There is only one prime ideal with norm 2, 3, or p^2 for p == 7, 11, 13, 17 (mod 24). Note that the norm of an element corresponding to a decomposing prime is p if p == 1, 19 (mod 24) and -p if p == 5, 23 (mod 24).
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
N(1 +- sqrt(6)) = -5, N(5 +- sqrt(6)) = 19, N(1 +- 2*sqrt(6)) = -23, N(5 +- 3*sqrt(6)) = -29, N(7 +- sqrt(6)) = 43, N(7 +- 4*sqrt(6)) = -47, ...
MATHEMATICA
seq[lim_] := Sort[Join[{2, 3}, Select[Range[lim], MemberQ[{1, 5, 19, 23}, Mod[#, 24]] && PrimeQ[#] &], Select[Range[Sqrt[lim]], MemberQ[{7, 11, 13, 17}, Mod[#, 24]] && PrimeQ[#] &]^2]]; seq[700] (* Amiram Eldar, Mar 30 2026 *)
PROG
(PARI) isA391371(n, {disc=24}) = (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A322796 ({kronecker(24,n)}), whose inverse Moebius transform A035188 gives the numbers of distinct ideals (or non-associate elements) with each norm (i.e., the coefficients of Dedekind zeta function).
Cf. A038876 (primes not inert in Q(sqrt(6))), A097934 (primes decomposing), A038877 (primes remaining inert).
Norms of prime ideals in the ring of integers of quadratic fields of class number 1: A391371 (D=24), A391370 (D=21), A391369 (D=12), A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341787 (D=-19), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163).
KEYWORD
nonn
AUTHOR
Jianing Song, Dec 07 2025
STATUS
approved
