Search: id:A056866
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%I A056866
%S A056866 60,120,168,180,240,300,336,360,420,480,504,540,600,660,672,720,780,
%T A056866 840,900,960,1008,1020,1080,1092,1140,1176,1200,1260,1320,1344,1380,
%U A056866 1440,1500,1512,1560,1620,1680,1740,1800,1848,1860,1920,1980,2016,2040
%N A056866 Orders of non-solvable groups, i.e. numbers which are not solvable numbers.
%C A056866 A number is solvable if every group of that order is solvable.
%C A056866 This comment is about the 4 sequences A001034, A060793, A056866, A056868: 
               The Feit Thompson theorem says that a finite group with odd order 
               is solvable, hence apart from the first trivial term of A060793 all 
               the other numbers in these sequences are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), 
               May 08 2001
%C A056866 Insoluble group orders can be derived from A001034 (simple non-cyclic 
               orders): n is an insoluble order iff n is a multiple of a simple 
               non-cyclic order - Des MacHale.
%H A056866 T. D. Noe, <a href="b056866.txt">Table of n, a(n) for n=1..2240</a> (orders 
               < 10^5)
%H A056866 J. Pakianathan and K. Shankar, <a href="http://www.math.ou.edu/~shankar/
               pubs.html">Nilpotent Numbers</a>, Amer. Math. Monthly, 107, August-September 
               2000, 631-634.
%H A056866 R. Brauer, <a href="http://www.pnas.org/cgi/reprint/47/12/1891.pdf">Investigation 
               On Groups Of Even Order, I</a>
%H A056866 R. Brauer, <a href="http://www.pnas.org/cgi/reprint/55/2/254.pdf">Investigation 
               On Groups Of Even Order, II</a>
%H A056866 W. Feit and J. G. Thompson, <a href="http://www.pnas.org/cgi/reprint/
               48/6/968.pdf">A Solvability Criterion For Finite Groups And Consequences</
               a>
%F A056866 A positive integer n is a non-solvable number if and only if it is a 
               multiple of any of the following numbers: a) 2^p(2^2p-1), p any prime. 
               b) 3^p(3^2p-1)/2, p odd prime. c) p(p^2-1)/2, p prime greater than 
               3 such that p^2+1 = 0 (mod 5). d) 2^4*3^3*13. e) 2^2p(2^2p+1)(2^p-1), 
               p odd prime.
%Y A056866 Cf. A003277, A051532, A056867, A056868, A001034.
%Y A056866 Sequence in context: A044001 A008887 A096490 this_sequence A098136 A060793 
               A087004
%Y A056866 Adjacent sequences: A056863 A056864 A056865 this_sequence A056867 A056868 
               A056869
%K A056866 nonn,easy,nice
%O A056866 1,1
%A A056866 N. J. A. Sloane (njas(AT)research.att.com), Sep 02 2000
%E A056866 More terms from Des MacHale (d.machale(AT)ucc.ie), Feb 19 2001
%E A056866 Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), 
               Dec 25 2001

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