A317990 -id:A317990 - cp's OEIS frontend

A317992 2-rank of the narrow class group of real quadratic field Q(sqrt(k)), k squarefree > 1.

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

Offset: 2

Jianing Song, Oct 03 2018

nonn

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.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Cf. A005117, A087048, A317990, A317991, A319662.

for(n=2, 200, if(issquarefree(n), print1(omega(n*if(n%4>1, 4, 1)) - 1, ", ")))

a(n) = omega(A005117(n)) - 1 = log_2(A317990(n)), where omega(k) is the number of distinct prime divisors of k.

Offset corrected by Jianing Song, Mar 31 2019

A317989 Number of genera of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 1, 4, 2, 2, 2, 2, 1, 4, 2, 2, 1, 2, 4, 1, 2, 4, 4, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 2, 2, 2, 2, 4, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 1, 4, 1, 4, 1, 2

Offset: 2

Jianing Song, Oct 03 2018

nonn

The number of genera of a quadratic field is equal to the number of elements x in the class group such that x^2 = e where e is the identity.

This is the analog of A003640 for real quadratic fields. Note that for this case "the class group" refers to the narrow class group, or the form class group of indefinite binary quadratic forms with discriminant k.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Cf. A003640, A003658, A087048, A317990, A317991.

2^(PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]]==#&]] - 1) (* Jean-François Alcover, Jul 25 2019 *)

for(n=2, 1000, if(isfundamental(n), print1(2^(omega(n) - 1), ", ")))

for(n=2, 1000, if(isfundamental(n), print1(2^#select(t->t%2==0, quadclassunit(n).cyc), ", ")))

def A317989_list(len):
    L = (sloane.A001221(n) for n in (1..len) if is_fundamental_discriminant(n))
    return [2^(l-1) for l in L]
A317989_list(290) # Peter Luschny, Oct 15 2018

a(n) = 2^(omega(A003658(n)-1)) = 2^A317991(n), where omega(k) is the number of distinct prime divisors of k.

Offset corrected by Jianing Song, Mar 31 2019

A338947 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at a vertex of one hexagon.

Original entry on oeis.org

1, 3, 6, 3, 6, 6, 6, 6, 3, 6, 12, 3, 6, 6, 6, 6, 6, 12, 6, 6, 9, 6, 12, 6, 12, 3, 6, 6, 6, 6, 6, 6, 12, 12, 12, 6, 3, 6, 6, 6, 12, 6, 12, 3, 6, 6, 12, 12, 6, 6, 18, 6, 6, 12, 6, 6, 9, 12, 6, 6, 6, 12, 12, 6, 6, 9, 6, 12, 6, 6, 12, 12, 6, 6, 12, 6, 12, 6, 6, 6, 12, 12, 3, 12, 6, 6, 24, 6, 12, 6, 3, 12, 6, 6, 12, 6, 6, 12

Offset: 0

Szymon Lukaszyk, Nov 17 2020

nonn

Radii of these circles are square roots of A003136.

This is A113062 with zeros dropped. - Andrey Zabolotskiy, Jun 21 2022

Szymon Lukaszyk, A short note about the geometry of graphene, ResearchGate, (2020).
Szymon Lukaszyk, Illustration of the subsequent 51 radii around a hexagon vertex.
S. Reich et al., Tight-binding description of graphene (up to third-nearest neighbors circles).

Cf. A003136, A113062; A338992 (similar but with circles centered at the center of one hexagon); A104888, A106030, A317990 (terms multiplied by 3 in agreement for up to 17 term).

a(0) changed from 0 to 1 by Andrey Zabolotskiy, Jun 21 2022

cp's OEIS Frontend

A317992 2-rank of the narrow class group of real quadratic field Q(sqrt(k)), k squarefree > 1.

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A317989 Number of genera of real quadratic field with discriminant A003658(n), n >= 2.

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A338947 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at a vertex of one hexagon.

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