OFFSET
1,1
COMMENTS
Also norms of prime ideals in Z[(1+sqrt(-163))/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 (-163,p) = (p,163) >= 0 and the squares of primes such that (-163,p) = (p,163) = -1, where (,) is the Legendre symbol.
For primes p such that (p,163) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-163))/2], namely (x + y*(1+sqrt(-163))/2) and (x + y*(1-sqrt(-163))/2), where (x,y) is a solution to x^2 + x*y + 41*y^2 = p; for p = 163, (sqrt(-163)) is the unique ideal with norm p; for primes p with (p,163) = -1, (p) is the only ideal with norm p^2.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
Some terms are N((1 +- sqrt(-163))/2) = 41, N((3 +- sqrt(-163))/2) = 43, N((5 +- sqrt(-163))/2) = 47, N((7 +- sqrt(-163))/2) = 53, ..., N((79 +- sqrt(-163))/2) = 1601.
PROG
(PARI) isA341783(n) = my(disc=-163); (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)
CROSSREFS
Cf. A011615 ({kronecker(-163,n)}), whose inverse Moebius transform A318983 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 norm are given by A318985.
Cf. A257362 (primes not inert in Q(sqrt(-163))), A296921 (primes decomposing), A296915 (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), this sequence (D=-163).
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Feb 19 2021
STATUS
approved