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A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.
(Formerly M3776)
13
1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

REFERENCES

H. Cohen, A Course in Computational Alg. No. Theory, Springer, 1993, pp. 515-519

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

Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.

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..3001

Steven R. Finch, Class number theory

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

Eric Weisstein's World of Mathematics, Dirichlet L-Series

Eric Weisstein's World of Mathematics, Fundamental Discriminant

Eric Weisstein's World of Mathematics, Class Number

FORMULA

Squarefree numbers (multiplied by 4 if not = 1 mod 4).

MATHEMATICA

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-Fran├žois Alcover, Nov 02 2011, after Eric Weisstein *)

Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)

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 *)

PROG

(PARI) v=[]; for(n=1, 500, if(isfundamental(n), v=concat(v, n))); v

CROSSREFS

Cf. A003657, A002144, A003646 (class numbers), A014000, A014046.

Sequence in context: A116602 A079896 A133315 * A003656 A003246 A143748

Adjacent sequences:  A003655 A003656 A003657 * A003659 A003660 A003661

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

EXTENSIONS

More terms from Eric W. Weisstein and Jason Earls (zevi_35711(AT)yahoo.com), Jun 19 2001

STATUS

approved

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Last modified October 21 06:15 EDT 2014. Contains 248375 sequences.