A003656 -id:A003656 - cp's OEIS frontend

A327297 Terms in A003656 that are not prime powers (A246655).

12, 21, 24, 28, 33, 44, 56, 57, 69, 76, 77, 88, 92, 93, 124, 129, 133, 141, 152, 161, 172, 177, 184, 188, 201, 209, 213, 217, 236, 237, 248, 249, 253, 268, 284, 301, 309, 329, 332, 341, 344, 376, 381, 393, 412, 413, 417, 428, 437, 453, 472, 489, 497, 501, 508, 517, 524, 536, 537, 553, 556, 573, 581, 589, 597

Offset: 1

Jianing Song, Sep 16 2019

nonn

Conjecture: if D is a term of this sequence, then D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4. For example, a(1) = 12 = 3*4, a(2) = 21 = 3*7, a(3) = 24 = 3*8, a(4) = 28 = 4*7, a(5) = 33 = 3*11, ... [This conjecture is correct: see Theorem 1 and Theorem 2 of Ezra Brown link; see also A003656. - Jianing Song, Dec 28 2021]
Let k be the quadratic field with discriminant D, O_k be ring of integers of k, N(x) be the norm of x and (D/p) be the Kronecker symbol. If D is a term of this sequence and D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4, then:
(a) if (((-u)/p), ((-v)/p)) = (1, 1), (1, 0) or (0, 1), then N(x) = p has solutions in O_k, while N(y) = -p has no solutions in k. For example, for D = 21 and p = 37, we have ((-3)/37) = ((-7)/37) = 1, and N(x) = 37 has solution x = (13 + sqrt(21))/2, but N(y) = -37 has no solutions in Q(sqrt(21)).
(b) if (((-u)/p), ((-v)/p)) = (-1, -1), (-1, 0) or (0, -1), then N(x) = -p has solutions in O_k, while N(y) = p has no solutions in k. For example, for D = 12 and p = 11, we have ((-3)/11) = ((-4)/11) = -1, and N(x) = -11 has solution x = 1 + 2*sqrt(3), but N(y) = 11 has no solutions in Q(sqrt(3)).
(c) if (((-u)/p), ((-v)/p)) = (1, -1) or (-1, 1), then N(x) = +-p has no solutions in k.
The smallest number of the form above that is not in this sequence is 316 = 4*79.
Also, it is conjectured that the quadratic field with discriminant D has form class number 2, where D is a term of this sequence. This is equivalent to the conjecture above. [This can also be deduced from the first paragraph of Ezra Brown link: the norm of the fundamental unit of the field k is -1 if D = 8 or a prime congruent to 1 modulo 4, and 1 if D is in this sequence. Here k is the quadratic field with discriminant D. - Jianing Song, Dec 28 2021]

Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.

Subsequence of A003656 and A003658.
Complement of A003655 with respect to A003656.
Cf. A016105, A246655.

isA327297(D) = if(D>1&&isfundamental(D), quadclassunit(D)[1]==1&&!isprimepower(D), 0)

A014539 Erroneous version of A003656.

Original entry on oeis.org

5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77

Offset: 0

dead

A071581 A003656(n)-prime(n).

Original entry on oeis.org

3, 5, 7, 6, 6, 8, 7, 9, 6, 4, 6, 4, 3, 10, 9, 4, 2, 8, 6, 5, 4, 9, 6, 3, -4, -4, -2, 2, 4, 11, 2, 2, 0, 2, 0, 1, 0, -2, 5, 0, -2, 0, -7, -5, -4, -2, -10, -14, -14, -12, 0, -3, -4, -10, -9, -14, -16, -3, -8, -4, -2, -9, -14, -10, -4, -4, -14, -8, -15, -12, -12, -15, -18, -20, -6, -7

Offset: 1

Benoit Cloitre, Jun 01 2002

sign

a(n)=0 for n = 33, 35, 37, 40, 42, 51 ...

More terms from Max Alekseyev, Jun 27 2011

A014602 Discriminants of imaginary quadratic fields with class number 1 (negated).

Original entry on oeis.org

3, 4, 7, 8, 11, 19, 43, 67, 163

Offset: 1

Eric Rains (rains(AT)caltech.edu)

Only fundamental discriminants are listed. The non-fundamental discriminants -12, -16, -27, and -28 also have class number 1 (and there are no others). - Andrew V. Sutherland, Apr 19 2009

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 229.
D. A. Cox, Primes of the form x^2+ny^2, Wiley, p. 271.
J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 483.

A. Abatzoglou, A. Silverberg, A. V. Sutherland, and A. Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, arXiv preprint arXiv:1404.0107 [math.NT], 2014.
Giacomo Cherubini and Alessandro Fazzari, Hyperbolic angles from Heegner points, arXiv:2206.08282 [math.NT], 2022. Mentions this sequence.
Charles Delorme and Guillermo Pineda-Villavicencio, Quadratic Form Representations via Generalized Continuants, Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.4.
Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Carl Ludwig Siegel, Zum Beweise des Starkschen Satzes, Inventiones mathematicae 5.3 (1968): 180-191.
Harold M. Stark, A complete determination of the complex quadratic fields of class-number one, The Michigan Mathematical Journal 14.1 (1967): 1-27.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656 (real case), A003173, A013658, A014603, A046002...A046020.

Union[ (-NumberFieldDiscriminant[ Sqrt[-#]] &) /@ Select[ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 &]] (* Jean-François Alcover, Jan 04 2012 *)

is(n)=isfundamental(-n) && qfbclassno(-n)==1 \\ Charles R Greathouse IV, Nov 20 2012

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
A014602 = lambda n: filter(is_fund_and_qfbcn_1, (-1,-2,..-n))
[-n for n in A014602(270)] # Peter Luschny, Aug 10 2014

A094612 Fundamental discriminants of real quadratic number fields with class number 3.

Original entry on oeis.org

229, 257, 316, 321, 469, 473, 568, 733, 761, 892, 993, 1016, 1101, 1229, 1257, 1304, 1373, 1436, 1489, 1509, 1772, 1901, 1929, 1957, 2021, 2089, 2101, 2177, 2213, 2429, 2557, 2589, 2636, 2677, 2713, 2777, 2857, 2917, 2981, 3173, 3221, 3229, 3261, 3356, 3569

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, A094613-A094614, A218156-A218160.

Select[Range[3569], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 3 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==3};
for(n=1, 3600, 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() == 3;
A094612 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094612(3700) # G. C. Greubel, Mar 01 2019

Edited by N. J. A. Sloane, May 01 2010

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 &]

A035120 Discriminants of real quadratic number fields with class number >= 2.

Original entry on oeis.org

40, 60, 65, 85, 104, 105, 120, 136, 140, 145, 156, 165, 168, 185, 204, 205, 220, 221, 229, 232, 257, 264, 265, 273, 280, 285, 296, 305, 312, 316, 321, 328, 345, 348, 357, 364, 365, 377, 380, 385, 401, 408, 424, 429, 440, 444, 445, 456, 460, 465, 469, 473

Offset: 1

N. J. A. Sloane

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.

T. D. Noe, Table of n, a(n) for n = 1..1000
X.-F. Roblot and Igor Schein, Hilbert class field of totally real fields of degree 2, 3 and 4.
Eric Weisstein's World of Mathematics, Class Number

Cf. A003656, A094619.

Select[Range[500], NumberFieldDiscriminant[Sqrt[#]] == # && NumberFieldClassNumber[Sqrt[#]] >= 2 & ] (* Jean-François Alcover, Jul 04 2013 *)

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

cp's OEIS Frontend

A327297 Terms in A003656 that are not prime powers (A246655).

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A014539 Erroneous version of A003656.

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A071581 A003656(n)-prime(n).

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A014602 Discriminants of imaginary quadratic fields with class number 1 (negated).

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A094612 Fundamental discriminants of real quadratic number fields with class number 3.

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

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Sage

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

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Sage

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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A035120 Discriminants of real quadratic number fields with class number >= 2.