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Given m the n-th cubefree number, A004709(n); a(n) is the class number of field Q(m^(1/3)).
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%I #56 Mar 23 2015 09:06:04

%S 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,

%T 3,12,1,1,1,2,3,1,3,3,1,1,6,6,1,3,6,3,6,18,6,6,3,1,9,1,3,3

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

%C The smallest m for which Q(m^(1/3)) not a unique factorization domain is m = 6, for which the corresponding field has class number 3.

%C 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 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 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 a(8) = 1 because the eighth cubefree number is 9 and Q(9^(1/3)) has class number 1.

%e a(9) = 2 because the ninth cubefree number is 10 and Q(10^(1/3)) has class number 2.

%Y Cf. A004709, A005472, A242867.

%K nonn,more

%O 1,6

%A _Alonso del Arte_, Aug 26 2014