A014000 -id:A014000 - cp's OEIS frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

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

Offset: 1

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

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

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).

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.

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

Union of A039955 and 4*A230375.

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 *)

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

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

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

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

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 8, 2, 1, 10, 3, 1, 4, 40, 1, 5, 2, 3, 250, 39, 1, 1, 42, 106, 5, 3, 1138, 2, 1, 8, 25, 146, 2, 273, 2968, 15, 6, 298, 1, 16, 2, 5, 6, 4, 17, 1856, 1, 2, 531, 1, 9384, 97, 3588, 10, 7, 253970, 2, 72664, 7, 3, 6440, 5, 521904, 12, 1, 1, 13

Offset: 2

Eric Rains (rains(AT)caltech.edu)

nonn

See A014000 for much more about this sequence. - N. J. A. Sloane, Jun 14 2013

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

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

Cf. A014000, A003658, A014077, A003652.

lista(nn) = { for (n=2, nn, if (isfundamental(n), nc = core(n); m = Mod (nc, 4); if ((m == 2) || (m == 3), d = 1); if ((m == 1), d = 4); b = 1; a = 0; while (a == 0, v = nc*b^2; if (issquare(v-d), a = sqrtint(v-d), if (issquare(v+d), a = sqrtint(v+d))); if (a == 0, b++; );); print1(b, ", ");););} \\ Michel Marcus, Sep 25 2018

Offset corrected by Jianing Song, Mar 31 2019

A014077 Norm of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

-1, -1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1

Offset: 2

Eric Rains (rains(AT)caltech.edu)

sign

See A014000 for details about the indexing.

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

Cf. A003658, A003652, A014000, A014046.

Offset corrected by Jianing Song, Mar 31 2019

A294176 Squarefree d such that the fundamental unit of Q(sqrt(d)) is larger than the fundamental unit of Q(sqrt(d)) for any smaller d.

Original entry on oeis.org

2, 3, 6, 7, 11, 14, 19, 22, 31, 43, 46, 67, 94, 127, 139, 151, 199, 211, 214, 331, 379, 454, 526, 571, 631, 739, 751, 886, 919, 991, 1291, 1366, 1699, 1726, 1999, 2011, 2311, 2326, 2566, 2671, 3019, 3259, 3691, 3931, 4174, 4846, 4951, 5119, 6211, 6379, 6406, 6451, 7606, 8254, 8779, 9619

Offset: 1

Alonso del Arte, Feb 10 2018

nonn

			The fundamental unit of Z[sqrt(2)] is 1 + sqrt(2) = 2.414213562373...
The fundamental unit of Z[sqrt(3)] is 2 + sqrt(3) = 3.7320508..., which is larger than 2.414213562373...
Thus the sequence starts out 2, 3.
The fundamental unit of O_(Q(sqrt(5))) is 1/2 + sqrt(5)/2 = 1.618..., which is actually smaller than the previous units, so 5 is not in the sequence.
The next term in the sequence is 6, corresponding to 5 + 2 sqrt(6) = 9.8989794855663561963945681494...

Robert G. Wilson v, Table of n, a(n) for n = 1..248

Cf. A014000, A014046.

k = 2; A294176 = {}; mx = 0; While[k < 1000, If[SquareFreeQ@ k && N[NumberFieldFundamentalUnits[Sqrt[k]], 16][[1]] > mx, mx = N[NumberFieldFundamentalUnits[Sqrt[k]], 16][[1]]; AppendTo[A294176, k]]; k++]; A294176 (* Robert G. Wilson v, Feb 11 2018 *)

a(10) onward from Robert G. Wilson v, Feb 11 2018

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A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

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A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A014077 Norm of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A294176 Squarefree d such that the fundamental unit of Q(sqrt(d)) is larger than the fundamental unit of Q(sqrt(d)) for any smaller d.

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