cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

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Keywords

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).

Crossrefs

Union of A039955 and 4*A230375.

Programs

  • 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 *)
  • 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

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

Extensions

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

A003656 Discriminants of real quadratic fields with unique factorization.

Original entry on oeis.org

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269
Offset: 1

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Comments

Discriminants of real quadratic fields with class number 1.
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

References

  • D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
  • H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
  • H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
  • Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A003652, A003658, A014602 (imaginary case).
For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

Programs

  • Mathematica
    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 *)
  • Sage
    is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
    A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
    A003656(270) # Peter Luschny, Aug 10 2014

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 8, 2, 19, 5, 3, 27, 10, 3, 15, 131, 4, 17, 7, 11, 943, 170, 4, 4, 197, 447, 24, 13, 5035, 9, 5, 37, 118, 703, 11, 1520, 15371, 79, 35, 1595, 6, 87, 11, 28, 37, 25, 98, 10847, 6, 13, 3482, 6, 57731, 604, 24335, 63, 48, 1637147, 13, 478763
Offset: 2

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Author

Eric Rains (rains(AT)caltech.edu)

Keywords

Comments

Taken from Cohen's table on pages 515-519. The table is indexed by the discriminant d = d(K) = A003658(n) of the real quadratic fields K. The fundamental unit is given as a pair of coordinates (a,b) = (A014000(n), A014046(n)) expressed in terms of the canonical integral basis (1,w) where w = (1+sqrt(d))/2 if d == 1 (mod 4), w = sqrt(d)/2 if d == 0 (mod 4).
The norm of this fundamental unit is A014077(n). The class number h(K) is A003652(n). - N. J. A. Sloane, Jun 14 2013

Examples

			Here is the start of Cohen's list of fundamental units: [0, 1], [1, 1], [2, 1], [1, 1], [3, 2], [2, 1], [5, 2], [8, 3], [2, 1], [19, 8], [5, 2], [3, 1], [27, 10], [10, 3], [3, 1], [15, 4], [131, 40],[4, 1], [17, 5], [7, 2], [11, 3], [943, 250], [170, 39], [4, 1], [4, 1], [197, 42], [447, 106], [24, 5], [13, 3], [5035, 1138], [9, 2], [5, 1], [37, 8], [118, 25], [703, 146], [11, 2], [1520, 273], [15371, 2968], [79, 15], [35, 6], [1595, 298], [6, 1], [87, 16], [11, 2], [28, 5], [37, 6], [25, 4], [98, 17], [10847, 1856], [6, 1], [13, 2], [3482, 531], [6, 1], [57731, 9384], [604, 97], [24335, 3588], [63, 10], [48, 7], [1637147, 253970], [13, 2], [478763, 72664], ... [_N. J. A. Sloane_, Jun 14 2013]
		

References

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

Crossrefs

Extensions

Edited by N. J. A. Sloane, Jun 14 2013
Offset corrected by Jianing Song, Mar 31 2019

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

Views

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

Comments

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

References

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

Crossrefs

Programs

  • PARI
    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

Extensions

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

Views

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

Comments

See A014000 for details about the indexing.

References

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

Crossrefs

Extensions

Offset corrected by Jianing Song, Mar 31 2019
Showing 1-5 of 5 results.