A005847 -id:A005847 - 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

A046085 Numbers n such that Q(sqrt(-n)) has class number 4.

Original entry on oeis.org

14, 17, 21, 30, 33, 34, 39, 42, 46, 55, 57, 70, 73, 78, 82, 85, 93, 97, 102, 130, 133, 142, 155, 177, 190, 193, 195, 203, 219, 253, 259, 291, 323, 355, 435, 483, 555, 595, 627, 667, 715, 723, 763, 795, 955, 1003, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555

Offset: 1

N. J. A. Sloane, Jun 16 2000

Contains 54 numbers [Arno, Theorem 7], ..., 1387, 1411, 1435, 1507 and 1555. [R. J. Mathar, May 01 2010]

Steven Arno, The imaginary quadratic fields of class number 4, Acta Arithm. vol 60 issue 4 (1991).
Steven Arno, M. L. Robinson, Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998) 295-330.
Keith Matthews, Tables of imaginary quadratic fields with small class numbers
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for sequences related to quadratic fields

See A003173, A005847, A006203, A046085, A046002, A055109, A046004, A055110, A046006, A055111 for class numbers 1 through 10.

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A055109 Numbers k such that Q(sqrt(-k)) has class number 6.

Original entry on oeis.org

26, 29, 38, 53, 61, 87, 106, 109, 118, 157, 202, 214, 247, 262, 277, 298, 339, 358, 397, 411, 451, 515, 707, 771, 835, 843, 1059, 1099, 1147, 1203, 1219, 1267, 1315, 1347, 1363, 1563, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787

Offset: 1

N. J. A. Sloane, Jun 16 2000

Jinyuan Wang, Table of n, a(n) for n = 1..51
Index entries for sequences related to quadratic fields

See A003173, A005847, A006203, A046085, A046002, A055109, A046004, A055110, A046006, A055111 for class numbers 1 through 10.

Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 6 &] (* Jinyuan Wang, Mar 08 2020 *)

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A055110 Numbers k such that Q(sqrt(-k)) has class number 8.

Original entry on oeis.org

41, 62, 65, 66, 69, 77, 94, 95, 105, 111, 113, 114, 137, 138, 141, 145, 154, 158, 165, 178, 183, 205, 210, 213, 217, 226, 238, 258, 265, 273, 282, 295, 299, 301, 310, 313, 322, 330, 337, 345, 357, 371, 382, 385, 395, 418, 438, 442, 445, 457

Offset: 1

N. J. A. Sloane, Jun 16 2000

Jinyuan Wang, Table of n, a(n) for n = 1..131
Index entries for sequences related to quadratic fields

See A003173, A005847, A006203, A046085, A046002, A055109, A046004, A055110, A046006, A055111 for class numbers 1 through 10.

Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 8 &] (* Jinyuan Wang, Mar 08 2020 *)

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A055111 Numbers k such that Q(sqrt(-k)) has class number 10.

Original entry on oeis.org

74, 86, 119, 122, 143, 159, 166, 181, 197, 218, 229, 303, 317, 319, 346, 373, 394, 415, 421, 422, 538, 541, 611, 613, 635, 694, 699, 709, 757, 779, 803, 851, 853, 877, 923, 982, 1093, 1115, 1213, 1318, 1643, 1707, 1779, 1819, 1835, 1891, 1923

Offset: 1

N. J. A. Sloane, Jun 16 2000

Jinyuan Wang, Table of n, a(n) for n = 1..87
Index entries for sequences related to quadratic fields

See A003173, A005847, A006203, A046085, A046002, A055109, A046004, A055110, A046006, A055111 for class numbers 1 through 10.

Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 10 &] (* Jinyuan Wang, Mar 08 2020 *)

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A191411 Class number, k, of n; i.e., imaginary quadratic fields negated Q(sqrt(-n))=k, or 0 if n is not squarefree (A005117).

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 1, 0, 2, 4, 2, 0, 4, 0, 1, 0, 4, 2, 3, 0, 0, 6, 0, 0, 6, 4, 3, 0, 4, 4, 2, 0, 2, 6, 4, 0, 8, 4, 1, 0, 0, 4, 5, 0, 0, 0, 2, 0, 6, 0, 4, 0, 4, 2, 3, 0, 6, 8, 0, 0, 8, 8, 1, 0, 8, 4, 7, 0, 4, 10, 0, 0, 8, 4, 5, 0, 0, 4, 3, 0, 4, 10, 6, 0, 12, 0, 2, 0, 4, 8, 8, 0, 4, 0, 0, 0, 14, 4, 5, 0, 8

Offset: 1

Robert G. Wilson v, Jun 01 2011

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

a(n)= 0: A013929; a(n)= 1: A003173; a(n)= 2: A005847; a(n)= 3: A006203; a(n)= 4: A046085; a(n)= 5: A046002; a(n)= 6: A055109; a(n)= 7: A046004; a(n)= 8: A055110; a(n)= 9: A046006; a(n)=10: A055111; a(n)=11: A046008; a(n)=12: n/a;
a(n)=13: A046010; a(n)=14: n/a; a(n)=15: A046012; a(n)=16: n/a; a(n)=17: A046014; a(n)=18: n/a; a(n)=19: A046016;
a(n)=20: n/a; a(n)=21: A046018; a(n)=22: n/a;
a(n)=23: A046020; a(n)=24: n/a; a(n)=25: A056987; etc.
Cf. A191408, A191410.
Cf. A000924 (without the zeros).

f[n_] := If[! SquareFreeQ@ n, 0, NumberFieldClassNumber@Sqrt@ -n]; Array[f, 105]

a(n) = if (! issquarefree(n), 0, qfbclassno(-n*if((-n)%4>1, 4, 1))); \\ Michel Marcus, Jul 08 2015

A236307 Discriminants d such that the ring of algebraic integers of Q(sqrt(-d)) is not a unique factorization domain.

Original entry on oeis.org

5, 6, 10, 13, 14, 15, 17, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Alonso del Arte, Apr 21 2014

nonn

Stewart & Tall (2002) show that none of the first thirteen terms listed here correspond to an imaginary quadratic ring with unique factorization by giving one example of an integer having two distinct factorizations for each ring.

This sequence consists of the squarefree numbers (A005117) that are not Heegner numbers (A003173).

			10 is in the sequence because 14 = 2 * 7 = (2 - sqrt(-10))(2 + sqrt(-10)), which are two distinct factorizations of 14 in Z[sqrt(-10)].
13 is in the sequence because 14 = 2 * 7 = (1 - sqrt(-13))(1 + sqrt(-13)), which are two distinct factorizations of 14 in Z[sqrt(-13)].
14 is in the sequence because 15 = 3 * 5 = (1 - sqrt(-14))(1 + sqrt(-14)), which are two distinct factorizations of 15 in Z[sqrt(-14)].
(Many more examples can be found for each ring; these three are from the thirteen given by Stewart & Tall (2002)).
And when -d = 1 mod 4 other than -3, -7, -11, -19, -43, -67 or -163, we can often use (d + 1)/4 = (1/2 - sqrt(-d)/2)(1/2 + sqrt(-d)/2) as an example, such as 4 = 2 * 2 = (1/2 - sqrt(-15)/2)(1/2 + sqrt(-15)/2) in O_(Q(sqrt(-15))).

Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): p. 83, Theorem 4.10.

Steven R. Finch, Class number theory, p. 5, Table 2. [Cached copy, with permission of the author]

Cf. A005117, A003173, A005847, A006203, A046085.

Select[Range[100], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[-#]] > 1 &]

a(n) = A005117(n + 9) for n > 91.

Name corrected after an e-mail from Michel Lagneau, Dec 25 2018

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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A046085 Numbers n such that Q(sqrt(-n)) has class number 4.

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A055109 Numbers k such that Q(sqrt(-k)) has class number 6.

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A055110 Numbers k such that Q(sqrt(-k)) has class number 8.

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A055111 Numbers k such that Q(sqrt(-k)) has class number 10.

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A191411 Class number, k, of n; i.e., imaginary quadratic fields negated Q(sqrt(-n))=k, or 0 if n is not squarefree (A005117).

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A236307 Discriminants d such that the ring of algebraic integers of Q(sqrt(-d)) is not a unique factorization domain.

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