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A119648
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Orders for which there is more than one simple group.
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2
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20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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).
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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]
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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]
Index entries for sequences related to groups
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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]
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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)
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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]
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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: A011786 A003801 A181233 * A127224 A114613 A052358
Adjacent sequences: A119645 A119646 A119647 * A119649 A119650 A119651
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jul 29 2006
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EXTENSIONS
| Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
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