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%I M3776 #80 Jan 21 2022 00:08:07
%S 1,5,8,12,13,17,21,24,28,29,33,37,40,41,44,53,56,57,60,61,65,69,73,76,
%T 77,85,88,89,92,93,97,101,104,105,109,113,120,124,129,133,136,137,140,
%U 141,145,149,152,156,157,161,165,168,172,173,177,181,184,185,188,193,197
%N Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.
%C All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - _Paul Muljadi_, Mar 28 2008
%C Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - _Charles R Greathouse IV_, Jan 21 2022
%D Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
%D M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
%D Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Amiram Eldar, <a href="/A003658/b003658.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..3001 from T. D. Noe)
%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a>, 2005. [Cached copy, with permission of the author]
%H Britta Habdank-Eichelsbacher, <a href="http://sfb343.math.uni-bielefeld.de/sfb343/preprints/pre95005.ps.gz">Unimodulare Gitter über Reell-Quadratischen Zahlkörpern</a>, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
%H Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FundamentalDiscriminant.html">Fundamental Discriminant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ClassNumber.html">Class Number</a>.
%F Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).
%F a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - _Charles R Greathouse IV_, Jan 21 2022
%t fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* _Jean-François Alcover_, Nov 02 2011, after _Eric W. Weisstein_ *)
%t Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &] (* _Alonso del Arte_, Apr 02 2014, based on _Arkadiusz Wesolowski_'s program for A094612 *)
%t max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* _Alonso del Arte_, Apr 02 2014 *)
%o (PARI) v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v
%o (PARI) list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ _Charles R Greathouse IV_, Jan 21 2022
%o (Sage)
%o def is_fundamental(d):
%o r = d % 4
%o if r > 1 : return False
%o if r == 1: return (d != 1) and is_squarefree(d)
%o q = d // 4
%o return is_squarefree(q) and (q % 4 > 1)
%o [1] + [n for n in (1..200) if is_fundamental(n)] # _Peter Luschny_, Oct 15 2018
%Y Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.
%Y Union of A039955 and 4*A230375.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Eric W. Weisstein_
%E More terms from _Eric W. Weisstein_ and _Jason Earls_, Jun 19 2001
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