A005472 -id:A005472 - cp's OEIS frontend

A005474 Class numbers of the real quadratic fields Q(sqrt(A005473(n))).

1, 1, 1, 1, 1, 3, 1, 3, 5, 3, 3, 7, 3, 5, 7, 3, 3, 5, 9, 7, 3, 5, 5, 15, 9, 19, 5, 13, 9, 9, 5, 19, 9, 5, 7, 15, 13, 9, 9, 15, 25, 13, 9, 27, 19, 15, 21, 7, 13, 11, 23, 9, 13, 13, 11, 33, 15, 25, 23, 15, 13, 29, 21, 17, 43, 35, 27, 33, 17, 17, 27, 45, 11, 63, 15, 31, 17, 15, 33, 15, 31, 31

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robin Visser, Table of n, a(n) for n = 1..10000
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152 (see Table 2 page 1143).
Index entries for sequences related to quadratic fields

Cf. A005472, A005473.

def a(n):
    m, k = 1, 1
    while (m < n): k += 1; m += (k^2+4).is_prime()
    return QuadraticField(k^2+4).class_number()  # Robin Visser, Dec 07 2024

More terms and name edited by Robin Visser, Dec 07 2024

A379019 Positive integers k such that the simplest cubic field defined by x^3 - kx^2 - (k+3)x - 1 is not monogenic.

Original entry on oeis.org

21, 30, 41, 48, 57, 75, 84, 90, 100, 102, 103, 111, 129, 138, 139, 152, 154, 156, 165, 183, 188, 192, 201, 204, 210, 219, 235, 237, 246, 250, 264, 269, 271, 273, 291, 299, 300, 318, 327, 335, 345, 348, 354, 356, 372, 374, 381, 384, 398, 399, 404, 408, 426, 433, 435, 438, 446, 453, 462, 480

Offset: 1

Robin Visser, Dec 13 2024

nonn

These are the positive integers k such that the ring of integers O_K of the simplest cubic field K = Q[x]/(x^3 - k*x^2 - (k+3)*x - 1) does not have a power integral basis of the form {1, a, a^2} for any element a in O_K.

D. Gil-Muñoz and M. Tinková, Additive structure of non-monogenic simplest cubic fields, arXiv:2212.00364 [math.NT], 2022.
T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.

Cf. A005472.

is_A379019 := function(k)
    R := PolynomialRing(Integers());
    K := NumberField(x^3 - k*x^2 - (k+3)*x - 1);
    return #IndexFormEquation(MaximalOrder(K), 1) eq 0;
end function;
[k : k in [1..1000] | is_A379019(k)];

A246457 Given m the n-th cubefree number, A004709(n); a(n) is the class number of the pure cubic field Q(m^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 3, 2, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 6, 1, 3, 12, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 6, 6, 1, 3, 6, 3, 6, 18, 6, 6, 3, 1, 9, 1, 3, 3, 1, 6, 3, 3, 6, 1, 2, 3, 3, 9, 1, 2, 3, 9, 3, 3, 3, 3, 3, 3, 1, 1, 2, 3, 3, 6, 6, 1, 3, 9, 3, 4, 3

Offset: 1

Alonso del Arte, Aug 26 2014

The smallest m for which the ring of integers of Q(m^(1/3)) is not a unique factorization domain is m = 7, for which the corresponding field has class number 3.

The table in Alaca & Williams includes 63 but excludes 18 and other cubefree but not squarefree numbers. It is clear that cubefree perfect squares are omitted from their table because on p. 328 they assert that Q((k^2)^(1/3)) = Q(k^(1/3)).

			a(8) = 1 because the eighth cubefree number is 9 and Q(9^(1/3)) has class number 1.
a(9) = 1 because the ninth cubefree number is 10 and Q(10^(1/3)) has class number 1.
a(10) = 2 because the tenth cubefree number is 11 and Q(11^(1/3)) has class number 2. - _Robin Visser_, Aug 31 2025

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 325-329, Examples 12.6.8 & 12.6.9, Table 9.

Robin Visser, Table of n, a(n) for n = 1..10000
Pierre Barrucand, H. C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976), no. 134, 312-323.
Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7-12.
Shin Nakano, Class numbers of pure cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 263-265.
Lawrence C. Washington, Class Numbers of the Simplest Cubic Fields, Mathematics of Computation, Vol. 48, No. 177 (January 1987): 371 - 384.

Cf. A000924, A003649, A004709, A005472, A242867.

def a(n):
    if n == 1: return 1
    m = [i for i in range(1, 2*n) if all([p[1]<3 for p in factor(i)])][n-1]
    K. = NumberField(x^3 - m)
    return K.class_number()  # Robin Visser, Aug 31 2025

Prepended a(1) = 1, corrected term a(43), and edited and more terms from Robin Visser, Aug 31 2025

cp's OEIS Frontend

A005474 Class numbers of the real quadratic fields Q(sqrt(A005473(n))).

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A379019 Positive integers k such that the simplest cubic field defined by x^3 - kx^2 - (k+3)x - 1 is not monogenic.

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A246457 Given m the n-th cubefree number, A004709(n); a(n) is the class number of the pure cubic field Q(m^(1/3)).

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cp's OEIS Frontend

A005474 Class numbers of the real quadratic fields Q(sqrt(A005473(n))).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Sage

Extensions

A379019 Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

A246457 Given m the n-th cubefree number, A004709(n); a(n) is the class number of the pure cubic field Q(m^(1/3)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

SageMath

Extensions

A379019 Positive integers k such that the simplest cubic field defined by x^3 - kx^2 - (k+3)x - 1 is not monogenic.