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%I M3777 #43 Mar 02 2021 04:40:56
%S 5,8,12,13,17,21,24,28,29,33,37,41,44,53,56,57,61,69,73,76,77,88,89,
%T 92,93,97,101,109,113,124,129,133,137,141,149,152,157,161,172,173,177,
%U 181,184,188,193,197,201,209,213,217,233,236,237,241,248,249,253,268,269
%N Discriminants of real quadratic fields with unique factorization.
%C Discriminants of real quadratic fields with class number 1.
%C Other than the term 8, every term is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. In fact, for a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of the form (i), (ii) or (iii). See Theorem 1 and Theorem 2 of Ezra Brown's link. - _Jianing Song_, Feb 24 2021
%D D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%D H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
%D H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
%D Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003656/b003656.txt">Table of n, a(n) for n = 1..1000</a>
%H Ezra Brown, <a href="https://doi.org/10.1090/S0002-9947-1974-0364172-9">Class numbers of real quadratic number fields</a>, Trans. Amer. Math. Soc. 190 (1974), 99-107.
%H Henri Cohen and X.-F. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01111-4">Computing the Hilbert Class Field of Real Quadratic Fields</a>, Math. Comp. 69 (2000), 1229-1244.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ClassNumber.html">Class Number</a>
%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%t 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-François Alcover_, Jan 24 2012 *)
%o (Sage)
%o is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
%o A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
%o A003656(270) # _Peter Luschny_, Aug 10 2014
%Y Cf. A003652, A003658, A014602 (imaginary case).
%Y For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002
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