A014602 -id:A014602 - cp's OEIS frontend

A351671 Discriminants of imaginary quadratic fields with class number 33 (negated).

839, 1583, 1951, 2423, 3967, 4091, 4423, 4567, 4663, 4831, 4999, 5167, 5623, 5791, 6343, 6823, 6967, 7331, 7351, 7499, 8167, 9011, 12619, 13183, 13619, 13931, 14251, 15299, 16619, 17419, 18691, 19163, 21347, 21563, 24019, 25411, 28027, 28163, 28579, 29243

Offset: 1

Andy Huchala, Mar 25 2022

Sequence contains 101 terms; largest is 222643.

The class group of Q[sqrt(-d)] is isomorphic to C_33 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..101
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

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] == 33]

A351673 Discriminants of imaginary quadratic fields with class number 35 (negated).

Original entry on oeis.org

1031, 1223, 2087, 2239, 2543, 4259, 4931, 5171, 5939, 6899, 7211, 7451, 7523, 8219, 8363, 8699, 9007, 9419, 10979, 11411, 11503, 12007, 14939, 15803, 16451, 16651, 17123, 18451, 19259, 20731, 22787, 23011, 24203, 24547, 26387, 26723, 28411, 33619, 36643

Offset: 1

Andy Huchala, Mar 25 2022

Sequence contains 103 terms; largest is 210907.

The class group of Q[sqrt(-d)] is isomorphic to C_35 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..103
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664.

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] == 35]

A351674 Discriminants of imaginary quadratic fields with class number 36 (negated).

Original entry on oeis.org

959, 1055, 1295, 1599, 1727, 1967, 2199, 2504, 2516, 2895, 3055, 3495, 3656, 3711, 3716, 3896, 3956, 4164, 4255, 4280, 4388, 4472, 4615, 4619, 4623, 4664, 4772, 5007, 5048, 5055, 5063, 5156, 5240, 5291, 5316, 5343, 5455, 5636, 5732, 5767, 5960, 6015, 6055

Offset: 1

Andy Huchala, Mar 27 2022

Sequence contains 668 terms; largest is 217627.

The class groups associated to 255 of the above discriminants are isomorphic to C_36, 374 have a class group isomorphic to C_18 X C_2, 16 have a class group isomorphic to C_12 X C_3, and the remaining 23 have a class group isomorphic to C_6 X C_6.

Andy Huchala, Table of n, a(n) for n = 1..668
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

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] == 36]

A351675 Discriminants of imaginary quadratic fields with class number 37 (negated).

Original entry on oeis.org

1487, 2447, 3391, 5839, 6367, 8147, 9803, 10739, 12343, 12583, 12967, 14767, 15259, 16927, 18947, 19403, 20011, 20147, 21139, 21587, 22807, 23371, 23627, 26731, 28283, 28307, 31699, 31723, 36691, 37171, 37243, 38371, 39139, 39451, 40531, 41659, 42283, 42443

Offset: 1

Andy Huchala, Mar 27 2022

Sequence contains 85 terms; largest is 158923.

The class group of Q[sqrt(-d)] is isomorphic to C_37 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..85
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

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] == 37]

A351676 Discriminants of imaginary quadratic fields with class number 38 (negated).

Original entry on oeis.org

1199, 1535, 1671, 2031, 3047, 3415, 4916, 5127, 5528, 6423, 6548, 6559, 6927, 7016, 7091, 7135, 7444, 8276, 8315, 8651, 8939, 8983, 9179, 9487, 9524, 9659, 9727, 9908, 10216, 10715, 10779, 10984, 11432, 11463, 11507, 11915, 12779, 12904, 13667, 14099, 14164

Offset: 1

Andy Huchala, Mar 27 2022

Sequence contains 237 terms; largest is 289963.

The class group of Q[sqrt(-d)] is isomorphic to C_38 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..237
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

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] == 38]

A351677 Discriminants of imaginary quadratic fields with class number 39 (negated).

Original entry on oeis.org

1439, 2207, 2791, 3767, 3919, 4111, 5099, 5119, 6199, 6779, 9059, 9967, 10091, 10163, 10399, 10567, 10667, 11743, 12539, 13163, 13523, 14843, 14867, 15607, 16087, 16139, 16787, 17383, 18127, 21851, 23027, 24499, 26539, 27827, 30211, 30347, 30803, 32027, 32491

Offset: 1

Andy Huchala, Mar 27 2022

Sequence contains 115 terms; largest is 253507.

The class group of Q[sqrt(-d)] is isomorphic to C_39 for all d in this sequence.

Andy Huchala, Table of n, a(n) for n = 1..115
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

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] == 39]

A351678 Discriminants of imaginary quadratic fields with class number 40 (negated).

Original entry on oeis.org

1271, 1839, 2255, 2415, 2559, 2751, 2756, 2919, 2936, 2959, 3044, 3135, 3255, 3399, 3423, 3524, 3704, 3927, 4004, 4047, 4071, 4407, 4607, 4760, 4807, 4820, 4836, 4856, 5060, 5143, 5191, 5304, 5367, 5727, 6020, 6036, 6212, 6324, 6807, 6980, 6996, 7063, 7080

Offset: 1

Andy Huchala, Mar 27 2022

Sequence contains 912 terms; largest is 260947.

The class groups associated to 251 of the above discriminants are isomorphic to C_40, 438 have a class group isomorphic to C_20 X C_2, and the remaining 223 have a class group isomorphic to C_10 X C_2 X C_2.

Andy Huchala, Table of n, a(n) for n = 1..912
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation, 73, pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number

Cf. A006203, A013658, A014602, A014603, A046002-A046020, A046125, A056987, A351664-A351680.

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] == 40]

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

T. D. Noe, May 19 2005

Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

			For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

Eric Weisstein's World of Mathematics, Class Number

Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).

A196923 Values of fundamental discriminant -d where number of ideal classes Q(sqrt(-d)) is at most two.

Original entry on oeis.org

3, 4, 7, 8, 11, 15, 19, 20, 24, 35, 40, 43, 51, 52, 67, 88, 91, 115, 123, 148, 163, 187, 232, 267, 403, 427

Offset: 1

Artur Jasinski, Oct 07 2011

Equals A014602 union A014603. - Michel Marcus, Nov 02 2013

A. Schinzel, Solution to a Problem of Lubelski and an Improvement of a Theorem of His, Bull. Pol. Acad. Sci. Math. Vol. 59 No 2 2011 pp. 115-119.
H. M. Stark, On complex quadratic fields with class number two, Math. Comp. 29, 1975, pp. 289-302.

A317970 Positive n such that the ring of algebraic integers O_n in a quadratic number field has class number 1 but is not Euclidean with respect to the norm.

Original entry on oeis.org

53, 56, 61, 69, 77, 88, 89, 92, 93

Offset: 1

N. J. A. Sloane, Aug 29 2018

The usual symbol in number theory is O_d, not O_n.

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.

Cf. A014602.

cp's OEIS Frontend

A351671 Discriminants of imaginary quadratic fields with class number 33 (negated).

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A351673 Discriminants of imaginary quadratic fields with class number 35 (negated).

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A351674 Discriminants of imaginary quadratic fields with class number 36 (negated).

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A351675 Discriminants of imaginary quadratic fields with class number 37 (negated).

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A351676 Discriminants of imaginary quadratic fields with class number 38 (negated).

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A351677 Discriminants of imaginary quadratic fields with class number 39 (negated).

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A351678 Discriminants of imaginary quadratic fields with class number 40 (negated).

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A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

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A196923 Values of fundamental discriminant -d where number of ideal classes Q(sqrt(-d)) is at most two.

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A317970 Positive n such that the ring of algebraic integers O_n in a quadratic number field has class number 1 but is not Euclidean with respect to the norm.