A014602 -id:A014602 - cp's OEIS frontend

A003173 Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).

1, 2, 3, 7, 11, 19, 43, 67, 163

Offset: 1

Could also be called Gauss numbers, since he discovered them. Heegner proved list is complete. - Artur Jasinski, Mar 21 2003
Numbers n such that Q(sqrt(-n)) has unique factorization into primes.
These are the squarefree values of n for which if some positive integer N can be written in the form (a/2)^2+n*(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n*(d/2)^2 for integers c and d. - V. Raman, Sep 17 2012, May 01 2013
Cases n = 1 and n = 2 correspond to the rings Z[i] (Gaussian integers) and Z[sqrt(-2)] = numbers of the form a + b*sqrt(-2), where a and b are integers. Other cases, satisfying a(n) == 3 (mod 4), correspond to the rings of numbers of the form (a/2) + (b/2)*sqrt(-a(n)), for integers a and b of the same parity. All these rings admit unique factorization. - V. Raman, Sep 17 2012, corrected by Eric M. Schmidt, Feb 17 2013
The Heegner numbers greater than 3 can also be found using the Kronecker symbol, as follows: A number k > 3 is a Heegner number if and only if s = Sum_{j = 1..k} j * (j|k) is prime, which happens to be negative, where (x|y) is the Kronecker symbol. Also note for these results s = -k. But if s = -k is used as the selection condition (instead of primality), then the cubes of {7, 11, 19, 43, 67, 163} are also selected, followed by these same numbers to 9th power (and presumably followed by the 27th or 81st power). - Richard R. Forberg, Jul 18 2016
Theorem: The ring of integers of the imaginary quadratic field Q(sqrt(-n)) is Euclidean iff n = 1, 2, 3, 7 and 11. (Otherwise, the ring of integers of the imaginary quadratic field Q(sqrt(-n)) is principal iff n is a term of this sequence) [Link Stark-Heegner theorem]. - Bernard Schott, Feb 07 2020
Named after the German high school teacher and radio engineer Kurt Heegner (1893-1965). - Amiram Eldar, Jun 15 2021

John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 224.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 213.
Wilfred W. J. Hulsbergen, Conjectures in Arithmetic Algebraic Geometry, Vieweg, 1994, p. 8.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 143.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Harold M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 295.

Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 9.
Kalyan Chakraborty, Azizul Hoque and Richa Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
Alex Clark and Brady Haran, 163 and Ramanujan Constant, Numberphile video, 2012.
Noam Elkies, The Klein quartic in number theory, in: S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999, pp. 51-101. MR1722413 (2001a:11103). See page 93.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Kurt Heegner, Diophantische Analysis und Modulfunktionen, Matematische Zaitschrift, Vol. 56 (1952), pp. 227-253.
John Myron Masley, Where are the number fields with small class number?, in: M. B. Nathanson (ed.), Number Theory Carbondale 1979, Lect. Notes Math., Vol. 751, Springer, Berlin, Heidelberg, 1982, pp. 221-242.
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, Gauss's Class Number Problem and Heegner Number.
Wikipedia, Heegner number.
Wikipedia, Stark-Heegner theorem.
Index entries for sequences related to quadratic fields

Cf. A003174, A005847 (for class number 2), A014602 (for discriminants of these fields), A048981, A263465.

Union[ Select[ -NumberFieldDiscriminant[ Sqrt[-#]]& /@ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 & ] /. {4 -> 1, 8 -> 2}] (* Jean-François Alcover, Jan 04 2012 *)
heegnerNums = {}; Do[s = Sum[j * KroneckerSymbol[j, k], {j, 1, k}]; If[PrimeQ[s], AppendTo[heegnerNums, {s, k}]], {k, 1, 10000}]; heegnerNums (* Richard R. Forberg, Jul 18 2016 *)

select(n->qfbclassno(-n*if(n%4==3,1,4))==1, vector(200,i,i)) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = A263465(n) = -A048981(6-n) for n <= 5. - Jonathan Sondow, May 28 2016

A006203 Discriminants of imaginary quadratic fields with class number 3 (negated).

Original entry on oeis.org

23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907

Offset: 1

N. J. A. Sloane

Also n such that Q(sqrt(-n)) has class number 3. Lubelski in 1936 proved that 907 is maximal term of this sequence. - Artur Jasinski, Oct 07 2011

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
J. M. Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998), pp. 295-330.
Kurt Heegner, Diophantische Analysis und Modulfunktionen, Matematische Zaitschrift, 1952, Vol. 56. p. 253.
S. Lubelski, Zur Reduzibilität von Polynomen in Kongruenzentheorie, Acta Arithmetica 1 (1935) pp. 169-183.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
Pieter Moree and Armand Noubissie, Higher Reciprocity Laws and Ternary Linear Recurrence Sequences, arXiv:2205.06685 [math.NT], 2022. See p. 4.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A005847, A013658, A014602, A014603, A046002, ..., A046020.
Cf. also A003173, A005847, ...
Cf. A191410.

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] & ) /@ Select[ Range[1000], NumberFieldClassNumber[ Sqrt[-#]] == 3 & ]] (* Jean-François Alcover, Jan 04 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 3} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..1000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==3] # G. C. Greubel, Mar 01 2019

A046002 Discriminants of imaginary quadratic fields with class number 5 (negated).

Original entry on oeis.org

47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683

Offset: 1

Eric W. Weisstein

Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046003-A046020.

Cf. A191410.

Union[(-NumberFieldDiscriminant[Sqrt[-#]] &) /@ Select[Range[2700], NumberFieldClassNumber[Sqrt[-#]] == 5 &]] (* Jean-François Alcover, Jun 27 2012 *)

select(n->qfbclassno(-n)==5,vector(670,n,4*n+3)) \\ Charles R Greathouse IV, Apr 25 2013

[n for n in (1..3000) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==5] # G. C. Greubel, Mar 01 2019

A014603 Discriminants of imaginary quadratic fields with class number 2 (negated).

Original entry on oeis.org

15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427

Offset: 1

Eric Rains (rains(AT)caltech.edu)

Includes only fundamental discriminants. The list of non-fundamental imaginary quadratic discriminants with class number 2 (negated) is 32, 36, 48, 60, 64, 72, 75, 99, 100, 112, 147. - Andrew V. Sutherland, Apr 08 2010

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 229.

A. Abatzoglou, A. Silverberg, A. V. Sutherland, A, Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, arXiv:1404.0107 [math.NT], 2014.
Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler, Principally Polarized Squares of Elliptic Curves with Field of Moduli Equal To Q, arXiv:1806.03826 [math.NT], 2018 (see table 1 page 4).
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, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A046002, ..., A046020.

Cf. A191410.

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] &) /@ Select[ Range[500], NumberFieldClassNumber[ Sqrt[-#]] == 2 &]] (* Jean-François Alcover, Jan 04 2012 *)

ok(n)={isfundamental(-n) && quadclassunit(-n).no == 2} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..500) if is_fundamental_discriminant(-n) and QuadraticField(-n, 'a').class_number()==2] # G. C. Greubel, Mar 01 2019

Offset corrected by Jianing Song, Aug 29 2018

A046020 Discriminants of imaginary quadratic fields with class number 23 (negated).

Original entry on oeis.org

647, 1039, 1103, 1279, 1447, 1471, 1811, 1979, 2411, 2671, 3491, 3539, 3847, 3923, 4211, 4783, 5387, 5507, 5531, 6563, 6659, 6703, 7043, 9587, 9931, 10867, 10883, 12203, 12739, 13099, 13187, 15307, 15451, 16267, 17203, 17851, 18379, 20323

Offset: 1

Eric W. Weisstein

Giovanni Resta, Table of n, a(n) for n = 1..68 (full sequence, from Steven Arno et al.)
Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields

Cf. A006203, A013658, A014602, A014603, A046002-A046020.

Reap[ Do[ If[ NumberFieldClassNumber[ Sqrt[-n] ] == 23, d = -NumberFieldDiscriminant[ Sqrt[-n] ]; Print[d]; Sow[d]], {n, 1, 21000}]][[2, 1]] // Union (* Jean-François Alcover, Oct 22 2012 *)

select(n->qfbclassno(-n)==23, vector(22696, n, 4*n+3)) \\ Charles R Greathouse IV, Apr 25 2013

68 discriminants in this sequence (proved).

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

A046125 Number of negative fundamental discriminants having class number n.

Original entry on oeis.org

9, 18, 16, 54, 25, 51, 31, 131, 34, 87, 41, 206, 37, 95, 68, 322, 45, 150, 47, 350, 85, 139, 68, 511, 95, 190, 93, 457, 83, 255, 73, 708, 101, 219, 103, 668, 85, 237, 115, 912, 109, 339, 106, 691, 154, 268, 107, 1365, 132, 345, 159, 770, 114, 427, 163, 1205, 179, 291

Offset: 1

Eric W. Weisstein

			a(1) = 9 because the discriminants {-3,-4,-7,-8,-11,-19,-43,-67,-163} are the only ones with class number 1.

Andy Huchala, Table of n, a(n) for n = 1..100 (from Watkins' paper)
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
K. Soundararajan, The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan Journal, Vol. 30 (2007), pp. 13-18.
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields.

Cf. A014602, A014603, A006203, A013658, A046002-A046016, A048925, A056987, A081319, A038552, A191408, A191410.

Cf. A038552, A306633.

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
k = 1; t = Table[0, {125}]; While[k < 2000001, If[ FundamentalDiscriminantQ@ -k, a = NumberFieldClassNumber@ Sqrt@ -k; If[a < 126, t[[a]]++]]; k++]; t (* Robert G. Wilson v Jun 01 2011 *)

lista(nn=10^7) = {my(NMAX=100, v = vector(NMAX), c); for (k=1, nn, if (isfundamental(-k), if ((c = qfbclassno(-k)) <= NMAX, v[c] ++););); v;} \\ Michel Marcus, Feb 17 2022

From Amiram Eldar, Apr 15 2025: (Start)
Formulas from Soundararajan (2007):
Sum_{k=1..n} a(k) = (3*zeta(2)/zeta(3)) * n^2 + O(n^2 * log(n)^(-1/2+eps)).
a(n) << n^2 * log(log(n))^4 / log(n). (End)

Edited by Robert G. Wilson v, May 13 2003

Corrected and extended by Dean Hickerson, May 20 2003. The values were obtained by transcribing and combining data from Tables 1-3 of Buell's paper, which has information for all values of n up to 125.

A056987 Discriminants of imaginary quadratic fields with class number 25 (negated).

Original entry on oeis.org

479, 599, 1367, 2887, 3851, 4787, 5023, 5503, 5843, 7187, 7283, 7307, 7411, 8011, 8179, 9227, 9923, 10099, 11059, 11131, 11243, 11867, 12211, 12379, 12451, 12979, 14011, 14923, 15619, 17483, 18211, 19267, 19699, 19891, 20347, 21107, 21323

Offset: 1

Eric W. Weisstein

Sequence contains 95 members; largest is 93307.

The class group of Q[sqrt(-d)] is isomorphic to C_5 X C_5 for d = 12451 and 37363. For all other d in this sequence, the class group of Q[sqrt(-d)] is isomorphic to C_25. - Jianing Song, Dec 01 2019

Jianing Song, Table of n, a(n) for n = 1..95
Eric Weisstein's World of Mathematics, Class Number.

Cf. A014602, A014603, A006203, A013658, A046002, A046003, A046004, A046005, A046006, A046007, A046008, A046009, A046010, A046011, A046012, A046013, A046014, A046015, A046016, A123563, A046018, A171724, A046020, A048925.

Reap[ For[n = 1, n < 22000, n++, s = Sqrt[-n]; If[ NumberFieldClassNumber[s] == 25, d = -NumberFieldDiscriminant[s]; Print[d]; Sow[d]]]][[2, 1]] // Union (* Jean-François Alcover, Oct 05 2012 *)

A351664 Discriminants of imaginary quadratic fields with class number 26 (negated).

Original entry on oeis.org

551, 951, 1247, 1256, 1735, 1832, 2651, 2771, 2792, 2823, 2839, 2984, 3092, 3327, 3368, 3611, 3736, 3903, 3992, 4052, 4207, 4244, 4376, 4739, 5123, 5435, 5524, 5891, 6059, 6443, 6515, 6587, 6676, 6847, 6891, 6964, 7156, 8003, 8339, 8383, 8408, 8691, 8743

Offset: 1

Andy Huchala, Feb 16 2022

Sequence contains 190 terms; largest is 103027.

The class group of Q[sqrt(-d)] is isomorphic to C_26 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..190
Eric Weisstein's World of Mathematics, Class Number.

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987.

ls = [(QuadraticField(-n, 'a').discriminant(), QuadraticField(-n, 'a').class_number()) for n in (0..10000) if is_fundamental_discriminant(-n) and not is_square(n)];
[-a[0] for a in ls if a[1] == 26]

A191410 Class number, k, of n, i.e.; imaginary quadratic fields negated Q(sqrt(-n))=k, or 0 if n is not a fundamental discriminant (A003657).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 4, 4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 3, 4, 0, 0, 6, 2, 0, 0, 2, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 5, 6, 0

Offset: 1

Robert G. Wilson v, Jun 01 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven Arno, M. L. Robinson and Ferrel S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arithm. 83.4 (1998), 295-330
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
C. Wagner, Class Number 5, 6 and 7, Math. Comput. 65, 785-800, 1996.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

a(n)= 0: n/a The complement of A003657; a(n)= 1: A014602; a(n)= 2: A014603; a(n)= 3: A006203; a(n)= 4: A013658; a(n)= 5: A046002; a(n)= 6: A046003; a(n)= 7: A046004; a(n)= 8: A046005; a(n)= 9: A046006; a(n)=10: A046007; a(n)=11: A046008; a(n)=12: A046009; a(n)=13: A046010; a(n)=14: A046011; a(n)=15: A046012; a(n)=16: A046013; a(n)=17: A046014; a(n)=18: A046015; a(n)=19: A046016; a(n)=20: A123563; a(n)=21: A046018; a(n)=22: A171724; a(n)=23: A046020; a(n)=24: A048925; a(n)=25: A056987; etc.

Cf. A003657, A191408.

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
f[n_] := If[ !FundamentalDiscriminantQ@ -n, 0, NumberFieldClassNumber@ Sqrt@ -n]; Array[f, 105]

a(n)=if(isfundamental(-n),qfbclassno(-n),0) \\ Charles R Greathouse IV, Nov 20 2012

cp's OEIS Frontend

A003173 Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).

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A006203 Discriminants of imaginary quadratic fields with class number 3 (negated).

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A046002 Discriminants of imaginary quadratic fields with class number 5 (negated).

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A014603 Discriminants of imaginary quadratic fields with class number 2 (negated).

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A046020 Discriminants of imaginary quadratic fields with class number 23 (negated).

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A003656 Discriminants of real quadratic fields with unique factorization.

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A046125 Number of negative fundamental discriminants having class number n.

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