A391437 2-rank of the class group of the real quadratic field Q(sqrt(k)), k squarefree > 1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2

Offset

2,79

Comments

Not to be confused with A317992, which gives the 2-ranks of the *form class groups* of real quadratic fields.

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 D = A005117(n), we have a(n) = A317992(n) - 1 if D has a prime factor congruent to 3 modulo 4, A317992(n) otherwise. (See A391439 for a proof). Note that

- A317992(n) = omega(A005117(n)) - 1 if A005117(n) == 1 (mod 4);

- A317992(n) = omega(4*A005117(n)) - 1 = omega(A005117(n)) - 1 if A005117(n) == 2 (mod 4);

- A317992(n) = omega(4*A005117(n)) - 1 = omega(A005117(n)) if A005117(n) == 3 (mod 4).

Prog

(PARI) r(D) = my(w = omega(D) - 1); my(f = factor(D)[, 1]~); for(i=1, #f, if(f[i]%4==3, return(w-1))); return(w) \\ gives 2-rank of Cl(D) for fundamental D
for(D=1, 200, if(D>1 && issquarefree(D), print1(r(D*if(D%4==1, 1, 4)), ", ")))

Crossrefs

Cf. A319662 (for imaginary quadratic fields).

For a list of sequences related to the class groups of real quadratic fields, see A390079.

Sequence in context: A063100 A353377 A294266 * A127475 A086014 A025437

Adjacent sequences: A391434 A391435 A391436 * A391438 A391439 A391440

Keyword

nonn

Author

Jianing Song, Dec 10 2025

Status

approved