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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)).

%I A246457 #65 Sep 01 2025 11:21:29
%S A246457 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,
%T A246457 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,
%U A246457 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
%N 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)).
%C A246457 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.
%C A246457 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)).
%D A246457 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.
%H A246457 Robin Visser, <a href="/A246457/b246457.txt">Table of n, a(n) for n = 1..10000</a>
%H A246457 Pierre Barrucand, H. C. Williams, and L. Baniuk, <a href="https://doi.org/10.2307/2005974">A computational technique for determining the class number of a pure cubic field</a>, Math. Comp. 30 (1976), no. 134, 312-323.
%H A246457 Taira Honda, <a href="https://doi.org/10.1016/0022-314X(71)90045-X">Pure cubic fields whose class numbers are multiples of three</a>, J. Number Theory 3 (1971), 7-12.
%H A246457 Shin Nakano, <a href="https://doi.org/10.3792/pjaa.59.263">Class numbers of pure cubic fields</a>, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 263-265.
%H A246457 Lawrence C. Washington, <a href="http://dx.doi.org/10.1090/S0025-5718-1987-0866122-8">Class Numbers of the Simplest Cubic Fields</a>, Mathematics of Computation, Vol. 48, No. 177 (January 1987): 371 - 384.
%e A246457 a(8) = 1 because the eighth cubefree number is 9 and Q(9^(1/3)) has class number 1.
%e A246457 a(9) = 1 because the ninth cubefree number is 10 and Q(10^(1/3)) has class number 1.
%e A246457 a(10) = 2 because the tenth cubefree number is 11 and Q(11^(1/3)) has class number 2. - _Robin Visser_, Aug 31 2025
%o A246457 (SageMath)
%o A246457 def a(n):
%o A246457     if n == 1: return 1
%o A246457     m = [i for i in range(1, 2*n) if all([p[1]<3 for p in factor(i)])][n-1]
%o A246457     K.<a> = NumberField(x^3 - m)
%o A246457     return K.class_number()  # _Robin Visser_, Aug 31 2025
%Y A246457 Cf. A000924, A003649, A004709, A005472, A242867.
%K A246457 nonn,changed
%O A246457 1,7
%A A246457 _Alonso del Arte_, Aug 26 2014
%E A246457 Prepended a(1) = 1, corrected term a(43), and edited and more terms from _Robin Visser_, Aug 31 2025