

A001034


Orders of noncyclic simple groups (without repetition).
(Formerly M5318 N2311)


25



60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920
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OFFSET

1,1


COMMENTS

An alternative definition, to assist in searching: Orders of noncyclic finite simple groups.
This comment is about the three sequences A001034, A060793, A056866: The FeitThompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even.  Ahmed Fares (ahmedfares(AT)mydeja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
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
The conjecture is not true. The smallest and the only counterexample among the first 457 terms of the sequence is a(175) = 138297600.
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)
Every term in this sequence is divisible by 4*p*q, where p and q are distinct odd primes.  Isaac Saffold, Oct 24 2021


REFERENCES

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
Dickson L.E. Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



CROSSREFS

Cf. A109379 (orders with repetition), A119648 (orders that are repeated).


KEYWORD

nonn,nice,core


AUTHOR



STATUS

approved



