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 A119648 Orders for which there is more than one simple group. 2
 20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS All such orders are composite numbers (since there is only one group of any prime order). Orders which are repeated in A109379. Contribution from Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). REFERENCES See A001034 for references and other links. C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] LINKS L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] David A. Madore, Orders of non-Abelian simple groups Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] FORMULA For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] EXAMPLE Contribution from Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start) For n=1 the a(1)=|A_8|=8!/2=20160, for n=2 the a(2)=|C_3(3)|=4585351680, for n=3 the a(3)=|C_3(5)|=228501000000000 and for n=4 the a(4)=|C_4(3)|=65784756654489600. (End) PROG (Other) sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable] CROSSREFS Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630. Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition). Sequence in context: A305186 A181233 A262776 * A127224 A235523 A235520 Adjacent sequences:  A119645 A119646 A119647 * A119649 A119650 A119651 KEYWORD nonn AUTHOR N. J. A. Sloane, Jul 29 2006 EXTENSIONS Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010 STATUS approved

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Last modified September 20 18:26 EDT 2018. Contains 315240 sequences. (Running on oeis4.)