OFFSET
1,1
COMMENTS
Also norms of prime ideals in Z[sqrt(3)], which is a unique factorization domain.
Consists of 2, 3, the primes p == 1, 11 (mod 12) and the squares of primes p such that p == 5, 7 (mod 12).
There are two distinct prime ideals with norm p == 1, 11 (mod 12), which decomposes in Q(sqrt(3)). There is only one prime ideal with norm 2, 3, or p^2 for p == 5, 7 (mod 12). Note that the norm of an element corresponding to a decomposing prime is p if p == 1 (mod 12) and -p if p == 11 (mod 12).
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
N(1 +- 2*sqrt(3)) = 11, N(5 +- 2*sqrt(3)) = 13, N(5 +- 4*sqrt(3)) = -23, N(7 +- 2*sqrt(3)) = 37, N(1 +- 4*sqrt(3)) = -47, ...
PROG
(PARI) isA391369(n, {disc=12}) = (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A110161 ({kronecker(12,n)}), whose inverse Moebius transform A035194 gives the numbers of distinct ideals (or non-associate elements) with each norm (i.e., the coefficients of Dedekind zeta function).
Cf. A038874 (primes not inert in Q(sqrt(3))), A097933 (primes decomposing), A003630 (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
