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
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.
Programs
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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
Comments