A391436 2-rank of the class group of the real quadratic field with discriminant A003658(n), n >= 2.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset

2,159

Comments

Not to be confused with A317991, 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 = A003658(n), we have a(n) = A317991(n) - 1 if D has a prime factor congruent to 3 modulo 4, A317991(n) otherwise. (See A391439 for a proof). Note that A317991(n) = omega(D) - 1.

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, 1000, if(D>1 && isfundamental(D), print1(r(D), ", ")))

Crossrefs

Cf. A319659 (for imaginary quadratic fields).

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

Sequence in context: A353472 A359475 A396865 * A294937 A389905 A363131

Adjacent sequences: A391433 A391434 A391435 * A391437 A391438 A391439

Keyword

nonn

Author

Jianing Song, Dec 09 2025

Status

approved