A003649 -id:A003649 - cp's OEIS frontend

A000924 Class number of Q(sqrt(-n)), n squarefree.

1, 1, 1, 2, 2, 1, 2, 1, 2, 4, 2, 4, 1, 4, 2, 3, 6, 6, 4, 3, 4, 4, 2, 2, 6, 4, 8, 4, 1, 4, 5, 2, 6, 4, 4, 2, 3, 6, 8, 8, 8, 1, 8, 4, 7, 4, 10, 8, 4, 5, 4, 3, 4, 10, 6, 12, 2, 4, 8, 8, 4, 14, 4, 5, 8, 6, 3, 6, 12, 8, 8, 8, 2, 6, 10, 10, 2, 5, 12, 4, 5, 4, 14, 8, 8, 3, 8, 4, 10, 8, 16, 14, 7, 8, 4, 6, 8, 10

Offset: 1

N. J. A. Sloane, Mira Bernstein

			a(10) = 4, since 14 is the 10th squarefree number and the class number of Q(sqrt(-14)) is 4.

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 322-325, Theorem 12.6.1, Example 12.6.6, Table 7.
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
R. A. Mollin, Quadratics, CRC Press, 1996, Appendix D, gives a table for n <= 1999, correcting that of Borevich and Shafarevich.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Index entries for sequences related to quadratic fields

Values of n run through A005117. Corresponding discriminants give A033197.

Cf. also A003649.

nmax = 100; s = Select[Range[2 * nmax], SquareFreeQ]; a[n_] := NumberFieldClassNumber[Sqrt[-s[[n]]]]; Table[a[n], {n, nmax}] (* Jean-François Alcover, Dec 30 2011 *)

lista(nn) = for (n=1, nn, if (issquarefree(n), print1(qfbclassno(-n*if((-n)%4>1, 4, 1)), ", "))); \\ Michel Marcus, Jul 08 2015

Edited by Dean Hickerson, Mar 17 2003

A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field).

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 29, 31, 33, 37, 38, 41, 43, 46, 47, 53, 57, 59, 61, 62, 67, 69, 71, 73, 77, 83, 86, 89, 93, 94, 97, 101, 103, 107, 109, 113, 118, 127, 129, 131, 133, 134, 137, 139, 141, 149, 151, 157, 158, 161, 163, 166, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201

Offset: 1

N. J. A. Sloane

Squarefree numbers n such that A003649 is 1. - T. D. Noe, Apr 02 2008

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 422-423.
E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields. British Association Mathematical Tables, Vol. 4, London, 1934. (See p. 1.)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 296.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 [terms 1 through 1000 by T. D. Noe]
A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974
R. G. Underwood, On the content bound for real quadratic field extensions, Axioms 2013, 2, 1-9; doi:10.3390/axioms2010001.
Index entries for sequences related to quadratic fields

Cf. A061574 (includes negative n), A029702-A029705, A218038-A218042.

Select[Range[2, 199], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[#]] == 1 &] (* Alonso del Arte, Apr 17 2015 *)

A007947(n)={my(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]); }
{ for (n=2, 10^3,
    if ( n!=A007947(n), next() );
    K = bnfinit(x^2 - n);
    if ( K.cyc == [], print1( n, ", ") );
); }
/* Joerg Arndt, Oct 18 2012 */

is(n)=issquarefree(n) && qfbclassno(if(n%4>1, 4, 1)*n)==1 \\ Charles R Greathouse IV, Jan 19 2017

The table in Borevich and Shafarevich extends to 497.

A076498 Class number h(p) of real quadratic field Q(sqrt(p)) where p is n-th prime == 1 mod 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Nov 11 2002

nonn

			h(p) = 1 for p < 229, h(229) = h(257) = h(733) = h(761) = 3, h(401) = 5 and h(577) = 7, ...

Herman te Riele and Hugh Williams, New computations concerning the Cohen-Lenstra heuristics, Experiment. Math. 12 (2003), no. 1, 99-113; alternative link.

Cf. A002144, A003649, A076499.

A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.

Original entry on oeis.org

10, 79, 82, 226, 730, 1534, 2305, 3601, 4762, 5626, 11026, 21610, 23410, 27226, 38026, 50626, 116554, 164026, 176401, 189226, 342226, 345745, 411394, 518401, 540226, 613090, 804610, 893026, 1071226, 1199026, 1299601, 1334026, 1550026, 2205226, 2433601, 2873026, 3515626, 3920401

Offset: 1

Emmanuel Vantieghem, Mar 13 2017

nonn

Every element d of the sequence is squarefree because, if f is the squarefree part of d, then Q(sqrt(f)) = Q(sqrt(d)). If f would be < d, the class number of Q(sqrt(f)) would not be < the class number of Q(sqrt(d)). Thus, f = d.

			The sequence starts with 10 because the class number of Q(sqrt(10)) = 2 and all fields Q(sqrt(m)) with m < 10 have class number 1.
The next term is 79 because the class number of Q(sqrt(79)) is 3 and all fields Q(sqrt(m)) with m < 79 have class number 1 or 2.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Robin Visser, Table of n, a(n) for n = 1..50

Cf. A003172, A003649, A283659.

A={}; hx = 1; d = 2; While[hx<300, d++; If[SquareFreeQ[d], h = NumberFieldClassNumber[Sqrt[d]]; If[h > hx, AppendTo[A,d]; hx = h]]]; A

classn(n) = qfbclassno(if(n%4>1, 4, 1)*n);
isok(d) = {if (issquarefree(d), cld = classn(d); for (k=2, d-1, if (issquarefree(k) && (classn(k) >= cld), return (0))); 1;);} \\ Michel Marcus, Mar 13 2017

More terms from Robin Visser, May 25 2024

A283659 Class numbers of the fields Q(sqrt(A283658(n))).

Original entry on oeis.org

2, 3, 4, 8, 12, 14, 16, 20, 22, 28, 44, 48, 52, 58, 74, 96, 116, 130, 153, 154, 176, 180, 200, 230, 240, 256, 288, 296, 312, 316, 357, 394, 412, 452, 504, 540, 574, 575, 584, 616, 692, 924, 994, 1061, 1068, 1080, 1245, 1248, 1302, 1336

Offset: 1

Emmanuel Vantieghem, Mar 13 2017

			The sequence starts with 2 because the first number in A283658 is 10 and the class number of Q(sqrt(10)) equals 2.
The fifth term is 12 because A283658(5) = 226 and the class number of Q(sqrt(226)) is 12.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Cf. A283658, A003649, A003172.

H = {}; hx = 1; d = 2; While[hx < 5, d++;
If[SquareFreeQ[d], h = NumberFieldClassNumber[Sqrt[d]];
  If[h > hx, AppendTo[H, h]; hx = h]]]; H

a(30)-a(50) from Robin Visser, May 25 2024

A246457 Given m the n-th cubefree number, A004709(n); a(n) is the class number of the pure cubic field Q(m^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 3, 2, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 6, 1, 3, 12, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 6, 6, 1, 3, 6, 3, 6, 18, 6, 6, 3, 1, 9, 1, 3, 3, 1, 6, 3, 3, 6, 1, 2, 3, 3, 9, 1, 2, 3, 9, 3, 3, 3, 3, 3, 3, 1, 1, 2, 3, 3, 6, 6, 1, 3, 9, 3, 4, 3

Offset: 1

Alonso del Arte, Aug 26 2014

The smallest m for which the ring of integers of Q(m^(1/3)) is not a unique factorization domain is m = 7, for which the corresponding field has class number 3.

The table in Alaca & Williams includes 63 but excludes 18 and other cubefree but not squarefree numbers. It is clear that cubefree perfect squares are omitted from their table because on p. 328 they assert that Q((k^2)^(1/3)) = Q(k^(1/3)).

			a(8) = 1 because the eighth cubefree number is 9 and Q(9^(1/3)) has class number 1.
a(9) = 1 because the ninth cubefree number is 10 and Q(10^(1/3)) has class number 1.
a(10) = 2 because the tenth cubefree number is 11 and Q(11^(1/3)) has class number 2. - _Robin Visser_, Aug 31 2025

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 325-329, Examples 12.6.8 & 12.6.9, Table 9.

Robin Visser, Table of n, a(n) for n = 1..10000
Pierre Barrucand, H. C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976), no. 134, 312-323.
Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7-12.
Shin Nakano, Class numbers of pure cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 263-265.
Lawrence C. Washington, Class Numbers of the Simplest Cubic Fields, Mathematics of Computation, Vol. 48, No. 177 (January 1987): 371 - 384.

Cf. A000924, A003649, A004709, A005472, A242867.

def a(n):
    if n == 1: return 1
    m = [i for i in range(1, 2*n) if all([p[1]<3 for p in factor(i)])][n-1]
    K. = NumberField(x^3 - m)
    return K.class_number()  # Robin Visser, Aug 31 2025

Prepended a(1) = 1, corrected term a(43), and edited and more terms from Robin Visser, Aug 31 2025

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A000924 Class number of Q(sqrt(-n)), n squarefree.

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A003172 Q(sqrt n) is a unique factorization domain (or simple quadratic field).

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A076498 Class number h(p) of real quadratic field Q(sqrt(p)) where p is n-th prime == 1 mod 4.

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A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.

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A283659 Class numbers of the fields Q(sqrt(A283658(n))).

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A246457 Given m the n-th cubefree number, A004709(n); a(n) is the class number of the pure cubic field Q(m^(1/3)).

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