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)).

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

Author

Alonso del Arte, Aug 26 2014

Keywords

• nonn
• changed

Comments

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)).

Examples

			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
		

References

• 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.

Links

• 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.

Crossrefs

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

Programs

SageMath

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

Extensions

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