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A003656 Discriminants of real quadratic fields with unique factorization.
(Formerly M3777)
21
5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Discriminants of real quadratic fields with class number 1.

REFERENCES

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.

Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.

Eric Weisstein's World of Mathematics, Class Number

Index entries for sequences related to quadratic fields

MATHEMATICA

maxDisc = 269; t = Table[ {NumberFieldDiscriminant[ Sqrt[n] ], NumberFieldClassNumber[ Sqrt[n] ]}, {n, Select[ Range[2, maxDisc], SquareFreeQ] } ]; Union[ Select[ t, #[[2]] == 1 && #[[1]] <= maxDisc & ][[All, 1]]] (* Jean-Fran├žois Alcover, Jan 24 2012 *)

PROG

(Sage)

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1

A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n))

A003656(270) # Peter Luschny, Aug 10 2014

CROSSREFS

Cf. A003652, A003658, A014602 (imaginary case).

For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

Sequence in context: A079896 A133315 A003658 * A003246 A143748 A124378

Adjacent sequences:  A003653 A003654 A003655 * A003657 A003658 A003659

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

STATUS

approved

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Last modified October 18 00:17 EDT 2019. Contains 328135 sequences. (Running on oeis4.)