A094612 -id:A094612 - 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

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

N. J. A. Sloane, Mira Bernstein

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

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).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

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.

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 *)

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

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

A094619 Fundamental discriminants of real quadratic number fields with class number 2.

Original entry on oeis.org

40, 60, 65, 85, 104, 105, 120, 136, 140, 156, 165, 168, 185, 204, 205, 220, 221, 232, 264, 265, 273, 280, 285, 296, 305, 312, 345, 348, 357, 364, 365, 377, 380, 385, 408, 424, 429, 440, 444, 456, 460, 465, 476, 481, 485, 488, 492, 493, 533, 545, 552, 561, 565

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094612-A094614, A218156-A218160.

Select[Range[565], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 2 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

is(n)=n>9 && isfundamental(n) && qfbclassno(n)==2 \\ Charles R Greathouse IV, Nov 05 2014

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 2;
A094619 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094619(600) # G. C. Greubel, Mar 01 2019

A094614 Fundamental discriminants of real quadratic number fields with class number 5.

Original entry on oeis.org

401, 817, 1093, 1393, 1429, 1641, 1756, 1897, 1996, 2081, 2153, 2908, 3121, 3181, 3253, 3928, 3997, 4353, 4357, 4441, 4504, 4757, 4889, 5241, 5269, 5308, 5944, 6113, 6157, 6481, 6949, 7064, 7229, 7513, 7817, 7861, 8049, 8108, 8121, 8309, 8501, 8689, 9181, 9229

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094613, A218156-A218160.

Select[Range[9229], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 5 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==5};
for(n=1, 10000, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 5;
A094614 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094614(10000) # G. C. Greubel, Mar 01 2019

A218156 Fundamental discriminants of real quadratic number fields with class number 6.

Original entry on oeis.org

697, 785, 940, 985, 1345, 1384, 1708, 1765, 1937, 2024, 2233, 2296, 2505, 2941, 2993, 3021, 3144, 3281, 3305, 3368, 3496, 3576, 3580, 3592, 3596, 3624, 3973, 4065, 4344, 4764, 4765, 4844, 5073, 5353, 5356, 5368, 5369, 5529, 5621, 5624, 5685, 5901, 6108, 6153

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Andy Huchala, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218157-A218160.

Select[Range[6153], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 6 &]

A218160 Fundamental discriminants of real quadratic number fields with class number 10.

Original entry on oeis.org

3129, 3585, 4097, 4321, 4444, 4865, 4904, 5777, 6085, 6945, 7049, 7221, 7705, 8124, 8321, 8569, 9321, 9340, 9448, 9553, 9669, 10408, 10885, 11281, 11509, 11921, 11937, 11944, 12984, 12993, 12997, 13516, 13741, 13865, 14392, 14396, 14529, 14745, 14888

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156-A218159.

Select[Range[14888], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 10 &]

A250235 Discriminants of real quadratic fields with cyclic 3-class group (3).

Original entry on oeis.org

229, 257, 316, 321, 469, 473, 568, 697, 733, 761, 785, 892, 940, 985, 993, 1016, 1101, 1229, 1257, 1304, 1345, 1373, 1384, 1436, 1489, 1509, 1708, 1765, 1772, 1901, 1929, 1937, 1957, 2021, 2024, 2089, 2101, 2177, 2213, 2233, 2296, 2429, 2505, 2557, 2589, 2636, 2677, 2713, 2777, 2857, 2917, 2920, 2941, 2981, 2993

Offset: 1

Daniel Constantin Mayer, Nov 14 2014

These real quadratic fields have class number divisible by 3 but not divisible by 9. Therefore, this sequence does not contain the discriminant 1129, since the corresponding quadratic field has cyclic 3-class group (9). However, this sequence contains the discriminant 697 whose corresponding quadratic field has class number 6=2*3. Note that 697 is not a member of the sequence A094612, where an exact class number 3 is required.
According to the Artin reciprocity law of class field theory, these real quadratic fields possess a cyclic cubic Hilbert 3-class field as their maximal unramified abelian 3-extension.
According to the Hasse formula d(K)=f^2*d for the discriminant d(K) of a non-Galois totally real cubic field in terms of the conductor f and the associated discriminant d of the real quadratic subfield of the normal closure of K, the sequence A006832 contains all discriminants d of real quadratic fields with class number divisible by 3, since they give rise to a totally real cubic field with conductor f=1 and discriminant d(K)=f^2*d=d. In particular, A006832 contains A250235.

Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. math. Sem. Univ. Hamburg 5 (1927), 353-363.
Helmut Hasse, Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage, Math. Z. 31 (1930), 565-582.
Helmut Hasse, Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage.

A094612 is a subsequence, A006832 is a supersequence.

for d := 2 to 3000 do a := false; if (1 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(d); C := ClassGroup(K); if ([3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;

A250236 Fundamental discriminants d such that the real quadratic field Q(sqrt(d)) and the complex quadratic field Q(sqrt(-3d)) both have cyclic 3-class groups of order 3.

Original entry on oeis.org

229, 257, 316, 321, 469, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1489, 1509, 1708, 1765, 1929, 1937, 2024, 2089, 2101, 2177, 2233, 2296, 2505, 2557, 2589, 2677, 2920, 2941, 2981, 2993

Offset: 1

Daniel Constantin Mayer, Nov 15 2014

nonn

Generally, the 3-class ranks s of the real quadratic field R=Q(sqrt(d)) and r of the complex quadratic field C=Q(sqrt(-3d)) are related by the inequalities s <= r <= s+1. This reflection theorem was proved by Scholz and independently by Reichardt using a combination of class field theory and Kummer theory over the bicyclic biquadratic compositum K=R*E of R with Eisenstein's cyclotomic field E=Q(sqrt(-3)) of third roots of unity.
In particular, the biquadratic field K=Q(sqrt(-3),sqrt(d)) has a 3-class group of type (3,3) if and only if s=r and R and C both have 3-class groups of type (3).
Therefore, the discriminants in the sequence A250236 uniquely characterize all complex biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.
The discriminant of K=R*E is given by d(K)=3^2*d^2 if gcd(3,d)=1 and simply by d(K)=d^2 if 3 divides d.

			A250236 is a proper subsequence of A250235. For instance, it does not contain the discriminant d=733, resp. 1373, although the corresponding real quadratic field R=Q(sqrt(d)) has 3-class group (3). The reason is that the 3-dual complex quadratic field C=Q(sqrt(-3d)) of R has 3-class group (9), resp. (27).

G. Eisenstein, Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzten Zahlen, J. Reine Angew. Math. 27 (1844), 289-310.

H. Reichardt, Arithmetische Theorie der kubischen Körper als Radikalkörper, Monatsh. Math. Phys. 40 (1933), 323-350.
A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math. 166 (1932), 201-203.

A250235 and A094612 are supersequences, A250237, A250238, A250239, A250240, A250241, A250242 are pairwise disjoint subsequences.

for d := 2 to 3000 do a := false; if (1 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then a := true; end if; end if; if (true eq a) then R := QuadraticField(d); E := QuadraticField(-3); K := Compositum(R,E); C := ClassGroup(K); if ([3,3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;

A094613 Fundamental discriminants of real quadratic number fields with class number 4.

Original entry on oeis.org

145, 328, 445, 505, 520, 680, 689, 777, 780, 793, 840, 876, 897, 901, 905, 924, 1020, 1045, 1096, 1105, 1145, 1160, 1164, 1221, 1288, 1292, 1313, 1320, 1365, 1480, 1560, 1640, 1677, 1736, 1740, 1745, 1752, 1820, 1848, 1885, 1932, 2005, 2040, 2056, 2120, 2145

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612, A094614, A218156-A218160.

Select[Range[2145], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 4 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==4};
for(n=1, 2500, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 4;
A094613 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094613(2500) # G. C. Greubel, Mar 01 2019

A081363 Smallest squarefree integer k such that Q(sqrt(k)) has class number n.

Original entry on oeis.org

2, 10, 79, 82, 401, 235, 577, 226, 1129, 1111, 1297, 730, 4759, 1534, 9871, 2305, 7054, 4954, 15409, 3601, 7057, 4762, 23593, 9634, 24859, 13321, 8761, 5626, 49281, 11665, 97753, 15130, 55339, 19882, 25601, 18226, 24337, 19834, 41614, 16899, 55966, 47959

Offset: 1

Dean Hickerson, Mar 19 2003

nonn

What is known about the asymptotics of this sequence? - Charles R Greathouse IV, Jan 26 2017

Records: 2, 10, 79, 82, 401, 577, 1129, 1297, 4759, 9871, 15409, 23593, 24859, 49281, 97753, 106537, 159199, 197137, 212137, 239119, 245023, 444089, 589822, 614849, 815413, 837929, 943951, 1025494, 1224121, 1240369, 1333255, 1334026, ..., . - Robert G. Wilson v, Apr 12 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..214

Cf. A081319, A081364, A094619, A094612, A094613, A094614.

Cf. A219361, A029702, A029703, A029704, A029705, A218038, A218039, A218040, A218041, A218042, A276541. - Robert G. Wilson v, Apr 12 2017

t[] := 0; k = 2; While[k < 56000, cn = NumberFieldClassNumber@Sqrt@k; If[ t[cn] == 0, t[cn] = k; Print[{cn, k}]]; k++]; t@# & /@ Range@ 42 (* _Robert G. Wilson v, Apr 12 2017 *)

v=vector(100); for(n=2,1e6, if(!issquarefree(n), next); t=qfbclassno(if(n%4>1,4*n,n)); if(t<=#v && v[t]==0, v[t]=n; print("a("t") = "n))) \\ Charles R Greathouse IV, Jan 26 2017

More terms from Max Alekseyev, Apr 28 2010

cp's OEIS Frontend

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

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A003656 Discriminants of real quadratic fields with unique factorization.

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A094619 Fundamental discriminants of real quadratic number fields with class number 2.

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A094614 Fundamental discriminants of real quadratic number fields with class number 5.

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A218156 Fundamental discriminants of real quadratic number fields with class number 6.

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A218160 Fundamental discriminants of real quadratic number fields with class number 10.

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A250235 Discriminants of real quadratic fields with cyclic 3-class group (3).

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A250236 Fundamental discriminants d such that the real quadratic field Q(sqrt(d)) and the complex quadratic field Q(sqrt(-3d)) both have cyclic 3-class groups of order 3.

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