This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A226142 #24 Dec 13 2015 07:18:02
%S A226142 1,1,2,3,3,4,4
%N A226142 The smallest positive integer k such that the symmetric group S_n is a product of k cyclic groups.
%C A226142 Since S_{n+1} is a product of a subgroup isomorphic to S_n and the cyclic group <(1,2,3,...,n+1)> we have a(n+1) <= a(n) + 1. On the other hand it is not clear that a(n) <= a(n+1) for all n. A lower bound is given by A226143(n) = ceiling(log(m)(n!)), m = A000793(n), a sequence that is not nondecreasing.
%C A226142 This sequence was suggested by a posting of _L. Edson Jeffery_ on the seqfans mailing list on May 24, 2013.
%C A226142 Cardinality of the smallest subset(s) X of S_n such that every permutation in S_n can be expressed as a product of some elements in X. - _Joerg Arndt_, Dec 13 2015
%H A226142 Miklós Abért, <a href="http://dx.doi.org/10.1112/S0024609302001042">Symmetric groups as products of Abelian subgroups</a>, Bull. Lond. Math. Soc., Volume 34, Issue 04, July 2002, pp. 451-456.
%e A226142 a(7) = 4 since a factorization of S_7 is given by C_1*C_2*C_3*C_4 where
%e A226142 C_1 = <(1,2,3,4)(5,6,7)>,
%e A226142 C_2 = <(1,4,6)(2,3,5,7)>,
%e A226142 C_3 = <(1,2,5,7)(3,4,6)>,
%e A226142 C_4 = <(1,3,5,6,7)(2,4)>,
%e A226142 and a brute force computation shows that S_7 is not a product of 3 or fewer cyclic subgroups.
%Y A226142 Cf. A051625, A225788.
%K A226142 nonn,hard,more,nice
%O A226142 1,3
%A A226142 _W. Edwin Clark_, May 27 2013