OFFSET
2,10
COMMENTS
The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the narrow class group of Q(sqrt(k)) or the form class group of indefinite binary quadratic forms with discriminant k, and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(k)) (cf. A317990).
This is the analog of A319662 for real quadratic fields.
Not to be confused with A391437, which gives the 2-ranks of the *class groups* of real quadratic fields. - Jianing Song, Dec 09 2025
LINKS
Jianing Song, Table of n, a(n) for n = 2..10000
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
FORMULA
For omega(k) the number of distinct prime divisors of k:
- a(n) = omega(4*A005117(n)) - 1 = omega(A005117(n)) if A005117(n) == 3 (mod 4). [Corrected by Jianing Song, Dec 10 2025]
MATHEMATICA
f[n_] := PrimeNu[n] - 1 + If[Mod[n, 4] > 1, Mod[n, 2], 0]; f /@ Select[Range[2, 200], SquareFreeQ] (* Amiram Eldar, Mar 28 2026 *)
PROG
(PARI) for(n=2, 200, if(issquarefree(n), print1(omega(n*if(n%4>1, 4, 1)) - 1, ", ")))
CROSSREFS
Sequences related to the class groups of real quadratic fields:
| Class groups | Form class groups |
-------------+-------------------------------+-------------------------------+
-------------+-------------------------------+-------------------------------+
For a list of sequences related to the class numbers of real quadratic fields, see A087048.
KEYWORD
nonn
AUTHOR
Jianing Song, Oct 03 2018
EXTENSIONS
Offset corrected by Jianing Song, Mar 31 2019
STATUS
approved
