A006641 -id:A006641 - cp's OEIS frontend

A191408 Duplicate of A006641.

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, 5, 2, 2, 4, 4, 3, 1, 4, 7, 5, 3, 4, 6, 2, 2, 8, 5, 6, 3, 8, 2, 6, 10, 4, 2, 5, 5, 4, 4, 3, 10, 2, 7, 6, 4, 10, 1, 8, 11, 4, 5, 8, 4, 2, 13, 4, 9, 4, 3, 6, 14, 4, 7, 5, 4, 12, 2, 2, 15, 6, 6, 8, 7, 12, 4, 8, 13, 8, 2, 11, 8, 4, 3, 14, 4, 4, 8, 10, 8

Offset: 1

Robert G. Wilson v, Jun 01 2011

dead

Same as A006641. - Georg Fischer, Oct 12 2018

Cf. A191410.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric W. Weisstein *);
NumberFieldClassNumber@ Sqrt@ # & /@ Select[-Range@ 300, FundamentalDiscriminantQ]

for(n=1, 300, if(isfundamental(-n), print1(quadclassunit(-n).no, ", "))) \\ Andrew Howroyd, Jul 23 2018

Class number of A003657(n).

Terms corrected by Andrew Howroyd and Robert G. Wilson v, Jul 24 2018

A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 3, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 5, 4, 0, 0, 2, 2, 0, 0, 3, 4, 0

Offset: 1

T. D. Noe, May 18 2005, Apr 30 2008

nonn

This sequence is closely related to the class number function, h(-n), which is given for fundamental discriminants in A006641. For a fundamental discriminant d, we have h(-d) < 2a(d). It appears that a(n) < Sqrt(n) for all n. For k>1, the primes p for which a(p)=k coincide with the numbers n such that the class number h(-n) is 2k-1 (see A006203, A046002, A046004, A046006. A046008, A046010, A046012, A046014, A046016 A046018, A046020). - T. D. Noe, May 07 2008

			a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.

See A106856.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A106856 (start of many quadratic forms).
Cf. A133675 (n such that a(n)=1).
Cf. A223708 (without zeros).

dLim=150; cnt=Table[0, {dLim}]; nn=Ceiling[dLim/4]; Do[d=b^2-4a*c; If[GCD[a, b, c]==1 && 0<-d<=dLim, cnt[[ -d]]++ ], {b, 0, nn}, {a, b, nn}, {c, a, nn}]; cnt

{a(n)=local(m); if(n<3, 0, forvec(v=vector(3,k,[0,(n+1)\4]), if( (gcd(v)==1)&(-v[1]^2+4*v[2]*v[3]==n), m++ ), 1); m)} /* Michael Somos, May 31 2005 */

A102264 Smallest prime for which 2^n exactly divides the class number h(-4p).

Original entry on oeis.org

5, 17, 41, 257, 521, 4481, 9521, 21929, 72089, 531977, 1256009, 5014169, 20879129, 70993529, 258844361, 866941841, 3771185921, 13949589209, 57388297721

Offset: 1

N. J. A. Sloane, Feb 19 2005

H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.
S. Louboutin, L-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field, Math. Comp. 59 (1992) 213-230, Table 1.
A. S. Mosunov and M. J. Jacobson Jr., Unconditional class group tabulation of imaginary quadratic fields to |Δ| < 2^40, Math. Comp. 85 (2016), no. 300, 1983-2009.

Cf. A006641.

def a(n):
    for p in Primes():
        if QuadraticField(-p).class_number().valuation(2)==n:
            return p  # Robin Visser, May 25 2024

a(10)-a(17) from Robin Visser, May 25 2024

a(18)-a(19) from Robin Visser, Dec 28 2024

A106031 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D < 0 is the n-th fundamental discriminant.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 3, 2, 1, 3, 4, 3, 2, 4, 4, 2, 2, 5, 3, 4, 2, 5, 2, 4, 6, 4, 2, 3, 3, 4, 3, 2, 6, 2, 4, 4, 3, 6, 1, 5, 6, 4, 3, 5, 3, 2, 7

Offset: 1

Steven Finch, May 05 2005

nonn

A006641 is the same except it is under the action of SL_2[Z].

			D = -3, -4, -7, -8, -11, -15, -19, -20, -23, -24, -31, ...,
that is, A003657 negated.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Jens Jonasson, Classes of integral binary quadratic forms, Master's thesis (2001), Appendix B.

Cf. A003657, A006641.

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A191408 Duplicate of A006641.

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A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

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A102264 Smallest prime for which 2^n exactly divides the class number h(-4p).

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A106031 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D < 0 is the n-th fundamental discriminant.

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