OFFSET
1,1
COMMENTS
Also norms of prime ideals in Z[(1+sqrt(21))/2], which is a unique factorization domain.
Consists of the primes such that (21,p) = (p,21) >= 0 and the squares of primes such that (p,21) = -1, where (,) is the kronecker symbol.
There are two distinct prime ideals with norm p == 1, 4, 5, 16, 17, 20 (mod 21), which decomposes in Q(sqrt(21)). There is only one prime ideal with norm 3, 7, or p^2 for p == 2, 8, 10, 11, 13, 19 (mod 21). Note that the norm of an element corresponding to a decomposing prime is p if p == 1, 4, 16 (mod 21) and -p if p == 5, 17, 20 (mod 21).
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
N((1 +- sqrt(21))/2) = -5, N(2 +- sqrt(21)) = -17, N((13 +- sqrt(21))/2) = 37, N((5 +- 3*sqrt(21))/2) = -41, N(8 +- sqrt(21)) = 43, N((1 +- 3*sqrt(21))/2) = -47, ...
PROG
(PARI) isA391370(n, {disc=21}) = (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A322829 ({kronecker(21,n)}), whose inverse Moebius transform A035203 gives the numbers of distinct ideals (or non-associate elements) with each norm (i.e., the coefficients of Dedekind zeta function).
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
