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 1-3 of 3 results.

A003655 Discriminants of real quadratic fields with narrow class number 1.

Original entry on oeis.org

5, 8, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 233, 241, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 593, 601, 613, 617, 641, 653, 661, 673
Offset: 1

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Or, positive fundamental discriminants with form class number 1.
All terms except 8 are primes congruent to 1 modulo 4. - Jianing Song, Jul 20 2022

References

  • D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Equals {8} U (A003656 intersect A002144).
Equals A003656 \ A327297.

Programs

  • PARI
    isA003655(n) = (n==8) || (isprime(n) && (n%4==1) && (qfbclassno(n)==1)) \\ Jianing Song, Jul 20 2022

Extensions

Better definition from David Brink, Dec 30 2007, Jan 01 2008

A349649 Discriminants of real quadratic fields whose fundamental unit has norm 1.

Original entry on oeis.org

12, 21, 24, 28, 33, 44, 56, 57, 60, 69, 76, 77, 88, 92, 93, 105, 120, 124, 129, 133, 136, 140, 141, 152, 156, 161, 165, 168, 172, 177, 184, 188, 201, 204, 205, 209, 213, 217, 220, 221, 236, 237, 248, 249, 253, 264, 268, 273, 280, 284, 285, 301, 305, 309, 312, 316
Offset: 1

Views

Author

Arkadiusz Wesolowski, Nov 23 2021

Keywords

References

  • D. A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 92-93.
  • Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 151.

Links

Crossrefs

Cf. A003658, A003653 (discriminants of real quadratic fields whose fundamental unit has norm -1).
Cf. A003656, A327297 (a subsequence).

Programs

  • PARI
    isok(D) = isfundamental(D) && norm(quadunit(D))==1;

A349419 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm 1.

Original entry on oeis.org

316, 321, 469, 473, 568, 817, 892, 993, 1016, 1101, 1257, 1304, 1393, 1436, 1509, 1641, 1756, 1761, 1772, 1897, 1929, 1957, 1996, 2021, 2101, 2177, 2429, 2589, 2636, 2908, 2913, 2981, 3173, 3261, 3356, 3569, 3736, 3873, 3928, 3941, 3957, 3981, 3997, 4009, 4193, 4281
Offset: 1

Views

Author

Jianing Song, Dec 29 2021

Keywords

Comments

Composite terms of A342368.
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 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. See Theorem 1 and Theorem 2 of Ezra Brown's link. This sequence gives values for d in the cases (ii) and (iii) and that the real quadratic number field with discriminant d has odd class number > 1.

Examples

			316 is a term since the quadratic field with discriminant 316 (Q(sqrt(79))) has class number 3. The fundamental unit of that field (80+9*sqrt(79)) has norm 1.
321 is a term since the quadratic field with discriminant 321 (Q(sqrt(321))) has class number 3. The fundamental unit of that field (215+12*sqrt(321)) has norm 1.

Links

Crossrefs

Intersection of A342368 and A349649. Equals A342368 \ A350165.
Cf. A003658, A003656, A327297.

Programs

  • PARI
    isA349419(D) = if(!isprime(D) && (D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
Showing 1-3 of 3 results.

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

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