A000924 -id:A000924 - cp's OEIS frontend

A003646 Class number of binary quadratic forms with fundamental discriminant A003658(n),n>=2.

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, 4, 1, 4, 2, 4, 1, 2, 4, 1, 2, 4, 4, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 4, 2, 2, 2, 4, 4, 3, 2, 1, 2, 2, 1, 2, 2, 2, 3, 4, 2, 2, 1, 4, 1, 4, 1, 2, 4, 1, 2, 2, 4, 2, 4, 1, 6, 1, 6, 4, 2, 2, 1, 2, 2, 4

Offset: 2

N. J. A. Sloane, Mira Bernstein

nonn

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 236, 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Wolfdieter Lang, Table for a(n), n=2, ..., 92 (see Part 2).

Cf. A087048, A079896.

A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 8, 2, 1, 10, 3, 1, 4, 40, 1, 5, 2, 3, 250, 39, 1, 1, 42, 106, 5, 3, 1138, 2, 1, 8, 25, 146, 2, 273, 2968, 15, 6, 298, 1, 16, 2, 5, 6, 4, 17, 1856, 1, 2, 531, 1, 9384, 97, 3588, 10, 7, 253970, 2, 72664, 7, 3, 6440, 5, 521904, 12, 1, 1, 13

Offset: 2

Eric Rains (rains(AT)caltech.edu)

nonn

See A014000 for much more about this sequence. - N. J. A. Sloane, Jun 14 2013

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A014000, A003658, A014077, A003652.

lista(nn) = { for (n=2, nn, if (isfundamental(n), nc = core(n); m = Mod (nc, 4); if ((m == 2) || (m == 3), d = 1); if ((m == 1), d = 4); b = 1; a = 0; while (a == 0, v = nc*b^2; if (issquare(v-d), a = sqrtint(v-d), if (issquare(v+d), a = sqrtint(v+d))); if (a == 0, b++; );); print1(b, ", ");););} \\ Michel Marcus, Sep 25 2018

Offset corrected by Jianing Song, Mar 31 2019

A014600 Class numbers h(D) of imaginary quadratic orders with discriminant D == 0 or 1 mod 4, D<0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 3, 2, 2, 2, 4, 2, 1, 3, 5, 2, 2, 2, 4, 4, 3, 2, 4, 2, 1, 4, 7, 2, 2, 3, 5, 4, 3, 4, 6, 2, 2, 3, 8, 4, 2, 2, 5, 6, 3, 3, 8, 2, 2, 6, 10, 4, 2, 3, 5, 4, 5, 4, 6, 4, 3, 6, 10, 4, 2, 2, 7, 6, 4, 4, 10, 4, 1, 8, 11, 4, 4, 3, 6, 6, 5, 4, 8, 4, 2, 5, 13, 4, 4

Offset: 0

Eric Rains (rains(AT)caltech.edu)

nonn

The sequence consists of class numbers of imaginary quadratic "orders", not imaginary quadratic "fields". The difference is that an imaginary quadratic order may be a non-maximal order, but a class number of an imaginary quadratic field always refers to the class number of the maximal order within that imaginary quadratic field. - David Jao, Sep 13 2020

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
William Cason, Akash Jim, Charlie Medlock, Erick Ross, Trevor Vilardi, and Hui Xue, Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace, arXiv:2407.11694 [math.NT], 2024. See p. 5.
Steven R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Erick Ross and Hui Xue, Signs of the Second Coefficients of Hecke Polynomials, arXiv:2407.10951 [math.NT], 2024. See p. 5.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

ClassList[n_?Negative] :=
Select[Flatten[#, 1] &@Table[
    {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],
  Mod[#3, 1] == 0 && #3 >= # &&
      GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]
a[n_] := Length[ClassList[Floor[n/2]*-4 - Mod[n,2] - 3]] (* David Jao, Sep 14 2020 *)

a(n)=qfbclassno(n\2*-4-n%2-3) \\ Charles R Greathouse IV, Apr 25 2013

a(n)=quadclassunit(n\2*-4-n%2-3).no \\ Charles R Greathouse IV, Apr 25 2013

Name corrected by David Jao, Sep 13 2020

A105389 Primes of the form x^2 + 32 y^2, also primes p with h(-p) divisible by 8.

Original entry on oeis.org

41, 113, 137, 257, 313, 337, 353, 409, 457, 521, 569, 577, 593, 761, 809, 857, 881, 953, 1129, 1153, 1201, 1217, 1249, 1321, 1553, 1601, 1657, 1777, 1889, 1993

Offset: 1

John L. Drost, May 01 2005

nonn

			41 = 9 + 32 * 1, 113 = 81 + 32 *1, 137 = 9 + 32*4

Barrucand, P. and Cohn, H. Note on primes of the form x^2 + 32 y^2, class number and residuacity, Journal fur die reine und angewandte Mathematik, v.238, pp. 67-70.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

QuadPrimes2[1, 0, 32, 10000] (* see A106856 *)
(* Second program: *)
max = 10^4; Table[yy = {y, 1, Floor[Sqrt[(max - x^2)/32]]}; Table[x^2 + 32 y^2, yy // Evaluate], {x, 1, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& (* Jean-François Alcover, Oct 04 2018 *)

A107998 Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form u + v*sqrt(m) for integer u, v.

Original entry on oeis.org

2, 3, 6, 7, 10, 11, 14, 15, 17, 19, 22, 23, 26, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 55, 57, 58, 59, 62, 65, 66, 67, 70, 71, 73, 74, 78, 79, 82, 83, 86, 87, 89, 91, 94, 95, 97, 101, 102, 103, 105, 106, 107, 110, 111, 113, 114, 115, 118, 119, 122, 123, 127

Offset: 1

Steven Finch, Jun 13 2005

nonn

H. C. Williams, Eisenstein's problem and continued fractions, Utilitas Math. 37 (1990) 145-157.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.
A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental unit

Cf. A107997.

Select[ Range[2, 127], (fu = NumberFieldFundamentalUnits @ Sqrt[#]; SquareFreeQ[#] && IntegerQ[fu[[1, 2, 1]] ] && IntegerQ[fu[[1, 2, 2]] ]) &] (* Jean-François Alcover, Jun 20 2013 *)

A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

Original entry on oeis.org

6, 7, 8, 2, 3, 4, 4, 9, 1, 9, 1, 7, 3, 9, 1, 9, 7, 8, 0, 3, 5, 5, 3, 8, 2, 7, 9, 4, 8, 2, 8, 9, 4, 8, 1, 4, 0, 9, 6, 3, 3, 2, 2, 3, 9, 1, 8, 9, 4, 4, 0, 1, 0, 3, 0, 3, 6, 4, 6, 0, 4, 1, 5, 9, 6, 4, 9, 8, 3, 3, 7, 0, 7, 4, 0, 1, 2, 3, 2, 3, 3, 2, 1, 3, 7, 6, 2, 1, 2, 2, 9, 3, 3, 4, 8, 4, 6, 1, 6, 8, 8, 8, 3, 2, 8

Offset: 0

R. J. Mathar, Aug 01 2010

Equals (49/64)*(668/729)*(15304/15625)*(116724/117649)*... inserting p= A000040 = 2, 3, 5, 7.. into the factor. Slightly larger than Product_{p=primes} (1-3/p^3) = 0.534566872085103888416775...

			0.678234491917391978035...

Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 85.
Takashi Taniguchi, A mean value theorem for the square of class number times regulator of quadratic extensions, Annales de l'institut Fourier. Vol. 58, No. 2 (2008), pp. 625-670; arXiv preprint, arXiv:math/0410531 [math.NT], 2004-2006.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Taniguchi's Constant.
Wikipedia, Euler Product.

Cf. A256392, A330523.

read("transforms") : efact := 1-3/p^3+2/p^4+1/p^5-1/p^6 ; Digits := 130 : tm := 310 : subs (p=1/x,1/efact) ; taylor(%,x=0,tm) : L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:

digits = 105; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s];
LR = LinearRecurrence[{0, 0, 3, -2, -1, 1}, {0, 0, -9, 8, 5, -33}, 2 m0];
r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-3/p^3+2/p^4+1/p^5-1/p^6) \\ Amiram Eldar, Mar 04 2021

More digits from Jean-François Alcover, Apr 15 2016

A346419 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Original entry on oeis.org

2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 2, 340, 1, 5, 394, 48, 10, 10, 4, 16, 5, 22, 3040, 2, 46, 70, 12, 12, 74, 50, 6, 64, 26, 6964, 20, 1, 48670, 96, 4, 2, 100, 3, 7, 10, 178, 30, 302, 198, 1060, 8, 39, 126, 16, 16, 130, 97684, 8, 25, 502, 6960, 2, 2136, 86, 4, 340, 9, 106, 160, 1, 18, 164, 5, 9, 20810

Offset: 1

Sean A. Irvine

nonn

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A346420, A048941, A048942.

nonSquares = Select[Range[72], !IntegerQ[Sqrt[#]]& ]; 2*NumberFieldFundamentalUnits[Sqrt[#]][[1, 2, 1]] & /@ nonSquares (* Jean-François Alcover, Nov 09 2012 *)

for(n=1,30,if(!issquare(n),print1(abs(2*polcoeff(lift(bnfinit(x^2-n).fu[1]),0)),","))) /* Ralf Stephan, Sep 18 2013; updated by Michel Marcus, Jun 25 2020 */

A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Original entry on oeis.org

2, 2, 1, 4, 6, 2, 2, 6, 2, 1, 8, 2, 2, 2, 78, 1, 1, 84, 10, 4, 2, 2, 6, 1, 4, 546, 2, 8, 12, 2, 2, 12, 8, 2, 10, 4, 1062, 6, 1, 7176, 14, 2, 2, 14, 1, 1, 4, 24, 8, 40, 26, 138, 2, 5, 16, 6, 2, 16, 11934, 2, 3, 60, 826, 2, 250, 10, 2, 78, 1, 12, 18, 1, 2, 18, 1, 1, 2244, 6, 84

Offset: 1

Sean A. Irvine

nonn

The radical part is actually sqrt(A007913(A000037(n))) where A007913(m) is the squarefree part of m. - Michel Marcus, Jun 26 2020

How does this sequence differ from A048942? The definitions of both sequences are identical, but the second comment in A048942 states the terms differ from n = 14 onwards. - Felix Fröhlich, Jun 16 2022

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A007913, A346419, A048941, A048942.

f(n) = {if (issquare(n), return (0)); if (!issquarefree(n), m = core(n), m = n); my(u = abs(2*polcoeff(lift(bnfinit(x^2-m, 1).fu[1]), 0))); if (u^2==1, return (1)); if (u^2==4, return (sqrtint((u^2+4)/m));); if (u^2 < 4, return((u^2+4)/n)); my(v2 = [(u^2-4)/m, (u^2+4)/m]); sqrtint(vecmin(select(x->denominator(x)==1, v2)));}
lista(nn) = apply(f, select(x->!issquare(x), [1..nn])); \\ Michel Marcus, Jun 25 2020; corrected Jun 16 2022

A104888 Class number of binary quadratic forms with radicand A005117(n).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 1, 2, 4, 1, 2, 2, 2, 2, 2, 1, 4, 2, 2, 4, 4, 1, 2, 4, 1, 4, 2, 2, 2, 4, 1, 4, 2, 2, 2, 1, 2, 2, 4, 2, 2, 4, 2, 1, 2, 2, 4, 6, 4, 2, 2, 2, 4, 1, 4, 2, 2, 4, 1

Offset: 2

Steven Finch, May 03 2005

nonn

The fundamental discriminant D and the radicand m (which is squarefree) are related via D=m if m=1 (mod 4) and D=4*m if m=2,3 (mod 4).

Hua Loo Keng, Introduction to Number Theory, Springer-Verlag, 1982, pp. 465-472.

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. A003646, A087048.

A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 0, 3, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 2, 0, 2

Offset: 1

Jianing Song, Sep 25 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 class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003643).

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

Cf. A000924, A003643, A005117, A033197, A319659.

Real discriminant case: A317992.

PrimeNu[#*If[Mod[-#, 4]>1, 4, 1]] - 1& /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Aug 02 2019 *)

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

def A319662_list(len):
    L = []
    for n in (1..len):
        if is_squarefree(n):
            if (-n) % 4 > 1: n <<= 2
            L.append(sloane.A001221(n) - 1)
    return L
print(A319662_list(141)) # Peter Luschny, Oct 15 2018

a(n) = log_2(A003643(n)) = omega(A005117(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

cp's OEIS Frontend

A003646 Class number of binary quadratic forms with fundamental discriminant A003658(n),n>=2.

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A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A014600 Class numbers h(D) of imaginary quadratic orders with discriminant D == 0 or 1 mod 4, D<0.

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A105389 Primes of the form x^2 + 32 y^2, also primes p with h(-p) divisible by 8.

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A107998 Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form u + v*sqrt(m) for integer u, v.

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Mathematica

A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

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Maple

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A346419 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

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A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Views

Author

Keywords

Comments

Links

Crossrefs

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PARI