OFFSET
1,1
COMMENTS
Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.
From Jianing Song, Feb 20 2021: (Start)
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[sqrt(-5)] has class number 2.
For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.
For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.
For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)
REFERENCES
David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.
A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
From Jianing Song, Feb 20 2021: (Start)
Let |I| be the norm of an ideal I, then:
|(2, 1+sqrt(-5))| = 2;
|(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;
|(sqrt(-5))| = 5;
|(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;
|(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;
|(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;
|(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)
PROG
(PARI) isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n, &p)); if(!e || e>2, 0, bitxor(e-1, !!vecsearch(ms, p%20))); }; \\ Antti Karttunen, Feb 24 2020
CROSSREFS
Cf. A091728.
Cf. A289741 ({kronecker(-20,n)}), whose inverse Moebius transform A035170 gives the numbers of distinct ideals with each norm (i.e., the coefficients of Dedekind zeta function).
Cf. A240920 (primes not inert in Q(sqrt(-5))), A139513 (primes decomposing), A003626 (primes remaining inert).
Cf. A033205 ((p) is the product of two principal ideals), A106865 ((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
easy,nonn
AUTHOR
Paul Boddington, Feb 02 2004
EXTENSIONS
Offset corrected by Jianing Song, Feb 20 2021
STATUS
approved
