A056868 -id:A056868 - cp's OEIS frontend

A001034 Orders of noncyclic simple groups (without repetition).

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

Offset: 1

N. J. A. Sloane, Simon Plouffe

An alternative definition, to assist in searching: Orders of non-cyclic finite simple groups.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
The primitive elements are A257146. These are also the primitive elements of A056866. - Charles R Greathouse IV, Jan 19 2017
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
From M. Farrokhi D. G., Aug 11 2020: (Start)
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

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. 137-168 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).

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000
C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (page images), Dover, NY, 1958, p. 309.
M. Farrokhi D. G., A GAP function generating the smallest non-cyclic finite simple group order greater than a given number m.
Walter Feit and J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.
M. Hall Jr., Simple groups of order less than one million, J. Alg. 20 (1) (1972) 98-102
David A. Madore, More terms
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

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

A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

Original entry on oeis.org

60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1620, 1680, 1740, 1800, 1848, 1860, 1920, 1980, 2016, 2040

Offset: 1

N. J. A. Sloane, Sep 02 2000

A number is solvable if every group of that order is solvable.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
Insoluble group orders can be derived from A001034 (simple non-cyclic orders): k is an insoluble order iff k is a multiple of a simple non-cyclic order. - Des MacHale
All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012
Subsequence of A056868 and hence of A060652. - Charles R Greathouse IV, Apr 16 2015, updated Sep 11 2015
The primitive elements are A257146. Since the sum of the reciprocals of the terms of that sequence converges, this sequence has a natural density and so a(n) ~ k*n for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015
From Jianing Song, Apr 04 2022: (Start)
Burnside's p^a*q^b theorem says that a finite group whose order has at most 2 distinct prime factors is solvable, hence all terms have at least 3 distinct prime factors.
Terms not divisible by 12 are divisible by 320 and have at least 4 distinct prime factors (cf. A257391). (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from T. D. Noe)
R. Brauer, Investigation on groups of even order, I.
R. Brauer, Investigation on groups of even order, II.
P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685-692. See p. 685.
W. Feit and J. G. Thompson, A solvability criterion for finite groups and consequences, Proc. N. A. S. 48 (6) (1962) 968.
J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
Cindy Tsang, Qin Chao, On the solvability of regular subgroups in the holomorph of a finite solvable group, arXiv:1901.10636 [math.GR], 2019.
Index entries for sequences related to groups

Subsequence of A000977 and A056868.

Cf. A003277, A051532, A056867, A001034, A060652.

ma[n_] := For[k = 1, True, k++, p = Prime[k]; m = 2^p*(2^(2*p) - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mb[n_] := For[k = 2, True, k++, p = Prime[k]; m = 3^p*((3^(2*p) - 1)/2); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mc[n_] := For[k = 3, True, k++, p = Prime[k]; m = p*((p^2 - 1)/2); If[Mod[p^2 + 1, 5] == 0, If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]]; md[n_] := Mod[n, 2^4*3^3*13] == 0; me[n_] := For[k = 2, True, k++, p = Prime[k]; m = 2^(2*p)*(2^(2*p) + 1)*(2^p - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; notSolvableQ[n_] := OddQ[n] || ma[n] || mb[n] || mc[n] || md[n] || me[n]; Select[ Range[3000], notSolvableQ] (* Jean-François Alcover, Jun 14 2012, from formula *)

is(n)={
    if(n%5616==0,return(1));
    forprime(p=2,valuation(n,2),
        if(n%(4^p-1)==0, return(1))
    );
    forprime(p=3,valuation(n,3),
        if(n%(9^p\2)==0, return(1))
    );
    forprime(p=3,valuation(n,2)\2,
        if(n%((4^p+1)*(2^p-1))==0, return(1))
    );
    my(f=factor(n)[,1]);
    for(i=1,#f,
        if(f[i]>3 && f[i]%5>1 && f[i]%5<4 && n%(f[i]^2\2)==0, return(1))
    );
    0
}; \\ Charles R Greathouse IV, Sep 11 2012

A positive integer k is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p*(2^(2*p)-1), p any prime. b) 3^p*(3^(2*p)-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^(2*p)*(2^(2*p)+1)*(2^p-1), p odd prime.

More terms from Des MacHale, Feb 19 2001

Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A056867 Nilpotent numbers: n such that every group of order n is nilpotent.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 125, 127, 128, 131, 133, 135, 137, 139

Offset: 1

N. J. A. Sloane, Sep 02 2000

Contains exactly the numbers n for which gcd(n,|A153038(n)|)=1 [Pazderski]. - R. J. Mathar, Apr 03 2012

A group G of order m is nilpotent iff it has a quotient group of order m/d for each divisor d of m. - Des MacHale and Bernard Schott, Jul 15 2022

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören, Archiv Math. 10 (1) (1959) 331.
Wikipedia, Nilpotent group.

Complement of A056868.
Cf. A003277, A051532, A056866.
Cf. A027748, A124010, A173557.

IsNilpotentInt := function(n)
  local f, i, j; f := PrimePowersInt(n);
  for i in [1..Length(f)/2] do
    for j in [1..f[2*i]] do
      if Gcd(f[2*i-1]^j-1, n) > 1 then return false; fi;
    od;
  od;
  return true;
end;
Filtered([1..140], IsNilpotentInt); # Gheorghe Coserea, Dec 02 2017

A153038[1] = 1; A153038[n_] := (x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x); A056867 = Select[Range[140], GCD[#, Abs[A153038[#]]] == 1 &] (* Jean-François Alcover, May 15 2012, after R. J. Mathar *)

is(n)=my(f=factor(n));for(k=1,#f[,1], for(j=1,f[k,2], if(gcd(n, f[k,1]^j-1)>1, return(0)))); 1 \\ Charles R Greathouse IV, Sep 18 2012

n is in this sequence if p^k is not congruent to 1 mod q for any primes p and q dividing n such that p^e but not p^(e+1) divides n and k <= e. - Charles R Greathouse IV, Aug 27 2012

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A060652 Orders of non-Abelian groups: n such that some group of order n is non-Abelian.

Original entry on oeis.org

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 17 2001

nonn

Because of the existence of a non-Abelian dihedral group of order 2n for each n>2 all the even numbers >= 6 are in this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000

Complement of A051532.
Union of A056868 and A046099. - Reinhard Zumkeller, Jun 28 2013
Cf. A027748, A124010, A212793.

a060652 n = a060652_list !! (n-1)
a060652_list = filter h [1..] where
   h x = any (> 2) (map snd pfs) || any (== 1) pks where
     pks = [p ^ k `mod` q | (p,e) <- pfs, q <- map fst pfs, k <- [1..e]]
     pfs = zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Jun 28 2013

abelianQ[n_] := Module[{f, lf, p, e, v}, f = FactorInteger[n]; lf = Length[f]; p = f[[All, 1]]; e = f[[All, 2]]; If[AnyTrue[e, # > 2&], Return[False]]; v = p^e; For[i = 1, i <= lf, i++, For[j = i+1, j <= lf, j++, If[Mod[v[[i]], p[[j]]] == 1 || Mod[v[[j]], p[[i]]] == 1, Return[False]]]]; Return[True]];
Select[Range[200], !abelianQ[#]&] (* Jean-François Alcover, Jul 19 2022, after Charles R Greathouse IV *)

is(n)=my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(1), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(1)))); 0 \\ Charles R Greathouse IV, Apr 16 2015

Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei>2 for some i or pi^k = 1 (mod pj) for some i and j and 1 <= k <= ei. - T. D. Noe, Mar 25 2007

Equivalently: Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei>2 for some i or if pi^ei = 1 (mod pj) for some i and j. - Charles R Greathouse IV, Jan 09 2022

More terms from T. D. Noe, Mar 11 2007

A109379 Orders of non-cyclic simple groups (with repetition).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 29 2006

The first repetition is at 20160 (= 8!/2) and the first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1869-1942). - David Callan, Nov 21 2006

By the Feit-Thompson theorem, all terms in this sequence are even. - Robin Jones, Dec 25 2023

See A001034 for references and other links.

David A. Madore, Table of n, a(n) for n = 1..493 [taken from link below]
David A. Madore, More terms
John McKay, The non-abelian simple groups g, |g|<10^6 - character tables, Commun. Algebra 7 (1979) no. 13, 1407-1445.
Ida May Schottenfels, Two Non-Isomorphic Simple Groups of the Same Order 20,160, Annals of Math., 2nd Ser., Vol. 1, No. 1/4 (1899), pp. 147-152.
Wikipedia, Feit-Thompson theorem.
Index entries for sequences related to groups

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders without repetition), A119648 (orders that are repeated).

A060702 Orders of finite groups that have trivial center.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 30, 34, 36, 38, 39, 42, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 66, 68, 70, 72, 74, 75, 78, 80, 82, 84, 86, 90, 93, 94, 96, 98, 100, 102, 106, 108, 110, 111, 114, 116, 118, 120, 122, 126, 129, 130, 132, 134, 136

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

nonn

Apart from the first element 1 this is a subsequence of A056868 because a nilpotent group has nontrivial center. If n = 0 mod 6 or n >= 6 and n = 2 mod 4 then n is in this sequence.
If n >= 6 and n == 2 mod 4 then n is a member of the sequence because of the dihedral group of order 2(2k+1). In addition, if p is a prime and p == 1 mod 4 then n=4p is a member of the sequence; however, if p == 3 mod 4 and p>5, then n=4p is not a member of the sequence. Furthermore, if n=pq where p and q are distinct odd primes with pDes MacHale and Mossie Crowe, Jul 05 2005
This sequence is closed under multiplication. - Eric M. Schmidt, Aug 27 2012

			The symmetric group S_3 of order 6 has trivial center so 6 belongs to the sequence.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A056868, A059806.

For the corresponding numbers of centerless groups of these orders see A357900.

The old entry 89 was an error, since it is a prime. - Robert F. Bailey (robertb(AT)math.carleton.ca) and Brett Stevens (brett(AT)math.carleton.ca), Jul 16 2009

Sequence extended and corrected by Eric M. Schmidt, Aug 27 2012

A069209 Orders of non-Abelian Z-groups.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Apr 14 2002

nonn

Z-groups are groups in which all Sylow subgroups are cyclic. n belongs to this sequence iff n is divisible by two distinct primes p and q, such that p divides q-1. This sequence contains sequence A064899 and it is a subsequence of sequence A056868.

Numbers n such that there is more than one Z-group of order n. - Eric M. Schmidt, Sep 15 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Wikipedia, Z-group.

Cf. A064899, A056868, A247371.

filter:= proc(n) local F,p,q; F:= numtheory:-factorset(n);
  for p in F do if member(1,map(`modp`,F,p)) then return true fi od:
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2014

filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, AnyTrue[pp, MemberQ[pp, q_ /; Divisible[q - 1, #]]&]];
Select[Range[2, 200], filterQ] (* Jean-François Alcover, Jul 18 2020 *)

def is_A069209(n) : return any((q-1)%p==0 for p,q in Combinations(prime_divisors(n),2)) # Eric M. Schmidt, Sep 15 2014

Edited and extended by, and missing term 78 added by, Eric M. Schmidt, Sep 15 2014

A131932 Number of nonisomorphic nonsolvable groups of order A056866(n).

Original entry on oeis.org

1, 3, 1, 1, 8, 1, 3, 6, 1, 26, 2, 2, 5, 2, 8, 23, 1, 6, 1, 107, 6, 1, 14, 1, 1, 1, 19, 2, 8, 28, 1, 93, 2, 4, 5, 5, 22, 1, 10, 1, 1, 588, 2, 20, 5, 1, 64, 4, 1, 5, 2, 5, 81, 1, 1, 18, 1, 25, 112, 2, 5, 1, 1

Offset: 1

Artur Jasinski, Jul 30 2007, Oct 20 2007

nonn

			a(1) = 1 because there is only 1 nonsolvable group of order 60: A_5 (alternating group of 5th degree).
a(2) = 3 because there are 3 different nonsolvable groups of order 120.

O. L. Hoelder, Bildung zusammengesetzter Gruppen, Math. Ann., 46 (1895), 321-422; see p. 420.

Cf. A001034, A060793, A056866, A056868.

NrUnsolvable := function(n) local i, count; count := 0; for i in [1..NumberSmallGroups(n)] do if not IsSolvableGroup(SmallGroup(n, i)) then count := count + 1; fi; od; return count; end; # Eric M. Schmidt, Apr 04 2013

LoadPackage("GrpConst"); NrUnsolvable := function(n) local i, j, num; num := 0; for i in DivisorsInt(n) do if i<>1 then for j in [1..NrPerfectGroups(i)] do num := num + Length(Remove(UpwardsExtensions(PerfectGroup(IsPermGroup, i, j), n/i))); od; fi; od; return num; end; # Eric M. Schmidt, Nov 14 2013

Edited by N. J. A. Sloane, Oct 08 2007
More terms from Eric M. Schmidt, Apr 04 2013
a(44)-a(63) from Eric M. Schmidt, Nov 14 2013

A068919 Squarefree orders of non-Abelian groups.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 30, 34, 38, 39, 42, 46, 55, 57, 58, 62, 66, 70, 74, 78, 82, 86, 93, 94, 102, 105, 106, 110, 111, 114, 118, 122, 129, 130, 134, 138, 142, 146, 154, 155, 158, 165, 166, 170, 174, 178, 182, 183, 186, 190, 194, 195, 201, 202, 203, 205, 206

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Mar 29 2002

nonn

This set is the difference A005117 \ A003277, that is, the squarefree numbers that are not cyclic (A003277). - Jeppe Stig Nielsen, Sep 25 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A060652 and A005117.
Cf. A064899.
Also intersection of A056868 and A005117. - Reinhard Zumkeller, Jun 28 2013
Cf. A008966.

a068919 n = a068919_list !! (n-1)
a068919_list = filter ((== 1) . a008966) a056868_list
-- Reinhard Zumkeller, Jun 28 2013

Complement[Select[Range[206], SquareFreeQ[#] &],
Select[Range[206], FiniteGroupCount[#] == 1 &]] (* Geoffrey Critzer, Nov 28 2015 *)

is(n)=issquarefree(n)&&gcd(n,eulerphi(n))!=1 \\ Jeppe Stig Nielsen, Sep 25 2017

A119648 Orders for which there is more than one simple group.

Original entry on oeis.org

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

All such orders are composite numbers (since there is only one group of any prime order).
Orders which are repeated in A109379.
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). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

			From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

See A001034 for references and other links.
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]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

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]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
W. 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]
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

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

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]

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. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

cp's OEIS Frontend

A001034 Orders of noncyclic simple groups (without repetition).

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A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

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A056867 Nilpotent numbers: n such that every group of order n is nilpotent.

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A060652 Orders of non-Abelian groups: n such that some group of order n is non-Abelian.

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Haskell

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PARI

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A109379 Orders of non-cyclic simple groups (with repetition).

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A060702 Orders of finite groups that have trivial center.

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A069209 Orders of non-Abelian Z-groups.

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Maple

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Sage

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A131932 Number of nonisomorphic nonsolvable groups of order A056866(n).

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