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%I M5318 N2311 #75 Jan 13 2023 12:10:26
%S 60,168,360,504,660,1092,2448,2520,3420,4080,5616,6048,6072,7800,7920,
%T 9828,12180,14880,20160,25308,25920,29120,32736,34440,39732,51888,
%U 58800,62400,74412,95040,102660,113460,126000,150348,175560,178920
%N Orders of noncyclic simple groups (without repetition).
%C An alternative definition, to assist in searching: Orders of non-cyclic finite simple groups.
%C This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by _Isaac Saffold_, Aug 09 2021]
%C The primitive elements are A257146. These are also the primitive elements of A056866. - _Charles R Greathouse IV_, Jan 19 2017
%C Conjecture: This is a subsequence of A083207 (Zumkeller numbers). Verified for n <= 156. A fast provisional test was used, based on Proposition 17 from Rao/Peng JNT paper (see A083207). For those few cases where the fast test failed (such as 2588772 and 11332452) the comprehensive (but much slower) test by _T. D. Noe_ at A083207 was used for result confirmation. - _Ivan N. Ianakiev_, Jan 11 2020
%C From _M. Farrokhi D. G._, Aug 11 2020: (Start)
%C The conjecture is not true. The smallest and the only counterexample among the first 457 terms of the sequence is a(175) = 138297600.
%C On the other hand, the orders of sporadic simple groups are Zumkeller. And with the exception of the smallest two orders 7920 and 95040, the odd part of the other orders are also Zumkeller. (End)
%C Every term in this sequence is divisible by 4*p*q, where p and q are distinct odd primes. - _Isaac Saffold_, Oct 24 2021
%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%D Dickson L.E. Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
%D M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H M. Farrokhi D. G., <a href="/A001034/b001034.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Cato, <a href="http://dx.doi.org/10.1090/S0025-5718-1977-0430052-X">The orders of the known simple groups as far as one trillion</a>, Math. Comp., 31 (1977), 574-577.
%H L. E. Dickson, <a href="https://hdl.handle.net/2027/coo.31924062612092">Linear Groups with an Exposition of the Galois Field Theory</a> (page images), Dover, NY, 1958, p. 309.
%H M. Farrokhi D. G., <a href="/A001034/a001034.txt">A GAP function generating the smallest non-cyclic finite simple group order greater than a given number m</a>.
%H Walter Feit and J. G. Thompson, <a href="https://doi.org/10.1073/pnas.48.6.968">A solvability criterion for finite groups and some consequences</a>, Proc. N. A. S. 48 (6) (1962) 968.
%H M. Hall Jr., <a href="https://doi.org/10.1016/0021-8693(72)90090-7">Simple groups of order less than one million</a>, J. Alg. 20 (1) (1972) 98-102
%H David A. Madore, <a href="http://www.madore.org/~david/math/simplegroups.html">More terms</a>
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%Y Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.
%Y Cf. A109379 (orders with repetition), A119648 (orders that are repeated).
%K nonn,nice,core
%O 1,1
%A _N. J. A. Sloane_, _Simon Plouffe_
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