OFFSET
1,1
COMMENTS
Let Cl+(D) and Cl(D) be the narrow class group and the class group of the quadratic order of discriminant D. Then Cl+(D)/Cl(D) = 1 if D < 0 or D > 0 and the fundamental unit has norm -1, C_2 if D > 0 and the fundamental unit has norm 1. Sequence gives D such that Cl+(D) = (C_2)^r for some r >= 0.
- If the fundamental unit has norm -1, then Cl+(D) = Cl(D), so D is in this sequence;
- If 16|D or D has a prime factor congruent to 3 modulo 4, then Cl+(D) = Cl(D) X C_2, so D is in this sequence;
- In the remaining cases, Cl+(D) has a cyclic subgroup of order 4, so D is not in this sequence.
See A391439 for more details.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
PROG
(PARI) isA390079(n) = if(n%4<=1 && !issquare(n) && !#select(k->k<>2, quadclassunit(n).cyc), if(n%16==0 || quadunitnorm(n)==-1, return(1)); my(f = factor(n)[, 1]~); for(i=1, #f, if(f[i]%4==3, return(1)))); return(0) \\ quadunitnorm() requires PARI-GP of version 2.15 or higher
CROSSREFS
Cf. A003171 (the sequence for imaginary quadratic fields).
Cf. A306638 (norms of the fundamental unit).
Sequences related to the class groups of real quadratic fields:
| Class groups | Form class groups |
-------------+-------------------------------+-------------------------------+
-------------+-------------------------------+-------------------------------+
| Exponent <= 2: A391419 | Exponent <= 2: this sequence |
For a list of sequences related to the class numbers of real quadratic fields, see A087048.
KEYWORD
nonn
AUTHOR
Jianing Song, Nov 24 2025
STATUS
approved
