OFFSET
1,1
COMMENTS
Consists of 2, 5, the primes p such that kronecker(40,p) = 1 and the squares of primes p such that kronecker(40,p) = -1.
Note that Q(sqrt(10)) has class number 2.
For primes p == 1, 9, 31, 39 (mod 40), there are two distinct ideals with norm p in Z[sqrt(10)], namely (x + y*sqrt(10)) and (x - y*sqrt(10)), where (x,y) is a solution to x^2 - 10*y^2 = +-p.
For p == 3, 13, 27, 37 (mod 40), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(10)) and (p, x-y*sqrt(10)), where (x,y) is a solution to x^2 - 10*y^2 = +-p^2 with y != 0; (2, sqrt(10)) and (3, sqrt(10)) are respectively the unique ideal with norm 2 and 5.
For kronecker(40,p) = -1, (p) is the only ideal with norm p^2.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
|(3, 1 +- sqrt(10))| = 3, |(13, 9 +- 5*sqrt(10))| = 13, |(3 +- 2*sqrt(10))| = 31, |(37, 39 +- 17*sqrt(10))| = 37, |(9 +- 2*sqrt(10))| = 41, |(43, 47 +- 6*sqrt(10))| = 43, ...
PROG
(PARI) isA391367(n, {disc=40}) = (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A391503 ({kronecker(40,n)}), whose inverse Moebius transform A035192 gives the numbers of distinct ideals with each norm (i.e., the coefficients of Dedekind zeta function).
Cf. A038879 (primes not inert in Q(sqrt(10))), A097955 (primes decomposing), A038880 (primes remaining inert).
Cf. A141180 ((p) is the product of two principal ideals), A141179 ((p) is the product of two non-principal ideals).
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,easy
AUTHOR
Jianing Song, Dec 07 2025
STATUS
approved
