A230375 -id:A230375 - 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

A230376 The left Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

Original entry on oeis.org

11, 1803647, 2699538733, 30778903, 112663560435723374699, 554945667652531, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 14710826638296122001733445931451

Offset: 1

Colin Barker, Oct 17 2013

nonn

The values of k are given by A005117, except for the leading 1.

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Apr 13 2022

			1803647 is in the sequence because it is an Aurifeuillian factor of 13^13-1.

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

Cf. A005117, A220983, A220984, A230375, A230377, A230378, A230379.

A230377 The left Aurifeuillian factor of k^k + 1 for k congruent to 0, 2 or 3 (mod 4) and squarefree.

Original entry on oeis.org

1, 1, 13, 113, 3541, 58367, 2826601, 19231, 113631466919, 9617835527609, 348275601426959, 35522826680397941, 241498479121, 8403855868042458448127, 1161044975606998832441701, 1272844589592126671, 10128165505710094110937686497, 4612290807753604561

Offset: 1

Colin Barker, Oct 17 2013

nonn

The values of k are given by A230375.

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Apr 25 2022

			58367 is in the sequence because it is an Aurifeuillian factor of 11^11+1.

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

Cf. A230375, A220983, A220984, A230375, A230376, A230378, A230379.

A230378 The right Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

Original entry on oeis.org

71, 13993643, 19152352117, 227633407, 813955076015309926319, 4098986195943739, 46959719470144429555105032871, 491873569944394295636860313807677, 1848593595048531176470116001230356265643249547, 108685909290746311448943506365699

Offset: 1

Colin Barker, Oct 17 2013

nonn

The values of k are given by A005117, except for the leading 1.

			13993643 is in the sequence because it is an Aurifeuillian factor of 13^13-1.

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.

Cf. A005117, A220983, A220984, A230375-A230377, A230376.

A230379 The right Aurifeuillian factor of k^k + 1 for k congruent to 0, 2 or 3 (mod 4) and squarefree.

Original entry on oeis.org

5, 7, 97, 911, 27961, 407353, 19955461, 142111, 870542161121, 73194743542229, 2498077661567473, 255982845098719961, 1784464680181, 63472256064447557254913, 8751868368432861971645641, 9227671631659104191, 73515350937824503550370503117, 34089603477374212561

Offset: 1

Colin Barker, Oct 17 2013

nonn

The values of k are given by A230375.

			407353 is in the sequence because it is an Aurifeuillian factor of 11^11+1.

Richard P. Brent, On computing factors of cyclotomic polynomials

A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

1, 75, 1, 16875, 24, 1, 221484375, 34560, 18, 1, 116279296875, 116121600, 58320, 39, 1, 12950606689453125, 780337152000, 440899200, 296595, 51, 1, 4861333986053466796875, 8899589151129600, 6666395904000, 68420017575, 663255, 63, 1, 677114376628875732421875

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
           1,            1,             1,              1,                 1
          75,           24,            18,             39,                51
       16875,        34560,         58320,         296595,            663255
   221484375,    116121600,     440899200,    68420017575,       20126472975
116279296875, 780337152000, 6666395904000, 93393323989875, 10382542981248375

Index entries for zeta function.

Cf. A370412 (numerators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 30, if(isfundamental(k), v=concat(v, denominator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, denominator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.denominator() # Robin Visser, Mar 19 2024

T(n, k) = denominator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = denominator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = denominator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 1, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

0, 2, 0, 4, 1, 0, 536, 11, 1, 0, 2888, 361, 23, 2, 0, 3302008, 24611, 1681, 116, 4, 0, 12724582576, 2873041, 257543, 267704, 328, 4, 0, 18194938976, 27233033477, 67637281, 3741352, 92656, 88, 1, 0, 875222833138832, 11779156811, 18752521534133, 1156377368, 479214352, 287536, 29, 2, 0

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
          0,           0,              0,               0,                 0
          2,           1,              1,               2,                 4
          4,          11,             23,             116,               328
        536,         361,           1681,          267704,             92656
       2888,       24611,         257543,         3741352,         479214352
    3302008,     2873041,       67637281,      1156377368,       14816172016
12724582576, 27233033477, 18752521534133, 753075777246704, 16476431095568992

Index entries for zeta function.

Cf. A370411 (denominators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 50, if(isfundamental(k), v=concat(v, numerator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, numerator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.numerator() # Robin Visser, Mar 19 2024

T(n, k) = numerator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = numerator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = numerator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 0, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

A072901 Composite numbers n such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n.

Original entry on oeis.org

6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 39, 42, 46, 51, 55, 58, 62, 66, 70, 74, 78, 82, 86, 87, 91, 94, 95, 102, 106, 110, 111, 114, 115, 118, 119, 122, 123, 130, 134, 138, 142, 143, 146, 154, 155, 158, 159, 166, 170, 174, 178, 182, 183, 186, 187, 190, 194, 195

Offset: 1

Benoit Cloitre, Aug 10 2002

Conjecture: All terms are squarefree. Example: 6=2*3; 15=3*5; 30=2*3*5; 154=2*7*11; 195=3*5*13. - Vincenzo Librandi, Aug 08 2010 and Michel Marcus, Nov 26 2013

If prime numbers were accepted, then sequence A230375 would have been obtained. - Michel Marcus, Nov 26 2013

Wikipedia, Quadratic field

Cf. A037449.

isok(n) = !isprime(n) && (quaddisc(n) == 4*n); \\ Michel Marcus, Nov 26 2013

cp's OEIS Frontend

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

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A230376 The left Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

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A230377 The left Aurifeuillian factor of k^k + 1 for k congruent to 0, 2 or 3 (mod 4) and squarefree.

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A230378 The right Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

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A230379 The right Aurifeuillian factor of k^k + 1 for k congruent to 0, 2 or 3 (mod 4) and squarefree.

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A370411 Square array T(n, k) = denominator( zeta_r(2*n) * sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Sage

Formula

A370412 Square array T(n, k) = numerator( zeta_r(2*n) * sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Sage

Formula

A072901 Composite numbers n such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).