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.

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

Views

Author

N. J. A. Sloane, Jun 16 2000

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 6 &] (* Jinyuan Wang, Mar 08 2020 *)
  • PARI
    \\  See A005847.

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

Views

Author

N. J. A. Sloane, Jun 16 2000

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 8 &] (* Jinyuan Wang, Mar 08 2020 *)
  • PARI
    \\ See A005847.

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

Views

Author

N. J. A. Sloane, Jun 16 2000

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[10000], MoebiusMu[#] != 0 && NumberFieldClassNumber[Sqrt[-#]] == 10 &] (* Jinyuan Wang, Mar 08 2020 *)
  • PARI
    \\ See A005847.

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

Views

Author

Robert G. Wilson v, Jun 01 2011

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Alonso del Arte, Apr 21 2014

Keywords

Comments

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).

Examples

			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))).

References

  • 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.

Links

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

Crossrefs

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

Programs

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

Formula

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

Extensions

Name corrected after an e-mail from Michel Lagneau, Dec 25 2018
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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