A003658 - OEIS

A003658

Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.
(Formerly M3776)

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

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

REFERENCES

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, 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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)

Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]

Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.

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)).

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

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-François Alcover, Nov 02 2011, after Eric W. 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

(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

(Sage)

def is_fundamental(d):

r = d % 4

if r > 1 : return False

if r == 1: return (d != 1) and is_squarefree(d)

q = d // 4

return is_squarefree(q) and (q % 4 > 1)

[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

CROSSREFS

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

Sequence in context: A116602 A079896 A133315 * A003656 A003246 A143748