

A341787


Norms of prime elements in Z[(1+sqrt(19))/2], the ring of integers of Q(sqrt(19)).


9



4, 5, 7, 9, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 169, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571
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OFFSET

1,1


COMMENTS

Also norms of prime ideals in Z[(1+sqrt(19))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,19) >= 0 and the squares of primes such that (p,19) = 1, where (p,19) is the Legendre symbol.
For primes p such that (p,19) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(19))/2], namely (x + y*(1+sqrt(19))/2) and (x + y*(1sqrt(19))/2), where (x,y) is a solution to x^2 + x*y + 5*y^2 = p; for p = 19, (sqrt(19)) is the unique ideal with norm p; for primes p with (p,19) = 1, (p) is the only ideal with norm p^2.


LINKS

Jianing Song, Table of n, a(n) for n = 1..10000


EXAMPLE

norm((1 + sqrt(19))/2) = norm((1  sqrt(19))/2) = 5;
norm((3 + sqrt(19))/2) = norm((3  sqrt(19))/2) = 7;
norm((5 + sqrt(19))/2) = norm((5  sqrt(19))/2) = 11;
norm((7 + sqrt(19))/2) = norm((7  sqrt(19))/2) = 17.


PROG

(PARI) isA341787(n) = my(disc=19); (isprime(n) && kronecker(disc, n)>=0)  (issquare(n, &n) && isprime(n) && kronecker(disc, n)==1)


CROSSREFS

Cf. A011585, A106863, A191019, A191063.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.
The total number of elements with norm n is given by A028641.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=3), A055025 (D=4), A090348 (D=7), A341784 (D=8), A341785 (D=11), A341786 (D=15*), this sequence (D=19), A091727 (D=20*), A341788 (D=43), A341789 (D=67), A341790 (D=163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
Sequence in context: A283554 A035243 A035255 * A335984 A286050 A047493
Adjacent sequences: A341784 A341785 A341786 * A341788 A341789 A341790


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Feb 19 2021


STATUS

approved



