OFFSET
1,1
COMMENTS
Also norms of prime ideals in Z[(1+sqrt(-11))/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 congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 11.
For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
N((1 +- sqrt(-11))/2) = 3, N((3 +- sqrt(-11))/2) = 5, N((9 +- sqrt(-11))/2) = 23, N((5 +- 3*sqrt(-11))/2) = 31, ...
PROG
(PARI) isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A011582 ({kronecker(-11,n)}), whose inverse Moebius transform A035179 gives the numbers of distinct ideals (or non-associate elements) with each norm (i.e., the coefficients of Dedekind zeta function).
The total numbers of elements with each norms are given by A028609.
Cf. A056874 (primes not inert in Q(sqrt(-11))), A296920 (primes decomposing), A191060 (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), this sequence (D=-11), A341787 (D=-19), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163).
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Feb 19 2021
STATUS
approved
