OFFSET
1,7
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)).
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.
EXAMPLE
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
PROG
(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.<a> = NumberField(x^3 - m)
return K.class_number() # Robin Visser, Aug 31 2025
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Alonso del Arte, Aug 26 2014
EXTENSIONS
Prepended a(1) = 1, corrected term a(43), and edited and more terms from Robin Visser, Aug 31 2025
STATUS
approved