A005180 -id:A005180 - cp's OEIS frontend

A002231 Primitive roots that go with the primes in A029932.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 43, 53, 79, 107, 149, 151, 163, 211, 223, 263, 277, 307, 347, 349, 367, 383, 461, 479, 503

Offset: 1

N. J. A. Sloane

Other known terms in the sequence: 541, 547, 617. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 18 2008

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.
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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XLV.

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Tomás Oliveira e Silva, Counts of least primitive roots of prime numbers (Artin's conjecture)
Tomás Oliveira e Silva, Least prime primitive roots
A. Paszkiewicz and A. Schinzel, On the least prime primitive root modulo a prime, Math. Comp. 71 (2002), no. 239, 1307-1321.
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

Cf. A029932.

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 18 2008

A327912 Orders of perfect non-simple groups.

Original entry on oeis.org

120, 336, 720, 960, 1080, 1320, 1344, 1920, 2160, 2184, 2688, 3000, 3600, 3840, 4860, 4896, 5040, 5376, 5760, 6840, 7200, 7500, 7560, 7680, 9720, 10080, 10752, 11520, 12144, 14400, 14520, 14580, 15000, 15120, 15360, 15600, 16464, 17280, 19656, 20160, 21504, 21600, 23040, 24360, 28224, 29160, 29760, 30240, 30720, 32256, 34560, 37500, 39600, 40320, 43008, 43200, 43320, 43740, 46080, 48000, 50616, 51840, 56448, 57600, 57624, 58240, 58320, 60480

Offset: 1

Sébastien Palcoux, Sep 29 2019

nonn

The smallest number n such that there is a simple group and a non-simple perfect group of order n is 20160. So this sequence is A060793 minus A001034 (as sets) for the orders less than 20160. The next known such exceptions are 181440, 262080, 443520 and 604800.

The perfect groups of order 61440, 122880, 172032, 245760, 344064, 491520, 688128, 983040 have not completely been determined yet. Then GAP neither provides the number of these groups nor the groups themselves.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.

The GAP Group, GAP Data Library "Perfect Groups".

Cf. A001034, A005180, A060793.

OrderPerfectNonSimple:=function(n1,n2)
   local it,S,G,L,o,No,i,c;
   it:=SimpleGroupsIterator(n1,n2);
   S:=[];
   for G in it do
      Add(S,Order(G));
   od;
   L:=[];
   for o in [n1..n2] do
      c:=0;
      for i in S do
         if i=o then
            c:=c+1;
         fi;
      od;
      No:=NumberPerfectGroups(o);
      if No>c then
         Add(L,o);
         if c>0 then
            Print([o,c,No]);
         fi;
      fi;
   od;
   return L;
end;;

A329191 The prime divisors of the orders of the sporadic finite simple groups.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 67, 71

Offset: 1

Hal M. Switkay, Nov 07 2019

This list is complete according to the classification theorem for finite simple groups.
This list includes all primes < 72 except 53 and 61, which do not divide the order of any sporadic finite simple group.
All entries on this list divide the order of the Monster, except 37, 43, and 67.

			The first term is necessarily 2, by the Feit-Thompson theorem.

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].

R. A. Wilson et al., ATLAS of Finite Group Representations, version 3, Sporadic groups

Cf. A001228, A005180, A174670.

A335419 Integers m such that every group of order m is not simple.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Bernard Schott, Jul 09 2020

nonn

Officially, the group of order 1 is not considered to be simple; "a group <> 1 is simple if it has no normal subgroups other than G and 1" (See reference for Joseph J. Rotman's definition).
There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.
As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.
The first composite even number that is not present in the data is 60 that is the order of simple alternating group Alt(5), the second one that is missing is 168 corresponding to simple Lie group PSL(3,2) [A031963].

			There exist 5 (nonisomorphic) groups of order 8: Z/8Z, Z/2Z × Z/4Z, (Z/2Z)^3, D_4 and H_8; none of these 5 groups is simple, so 8 is a term.
There exist 13 (nonisomorphic) groups of order 60 (see A000001), 12 are not simple but the alternating group Alt(5) is simple, hence 60 is not a term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.

Craig Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
Wikipedia, Feit-Thompson theorem.

Complement of A005180 (except for 1).
Subsequence: A014076 (odd nonprimes).
Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).

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A002231 Primitive roots that go with the primes in A029932.

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A327912 Orders of perfect non-simple groups.

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A329191 The prime divisors of the orders of the sporadic finite simple groups.

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A335419 Integers m such that every group of order m is not simple.

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