A056866 -id:A056866 - cp's OEIS frontend

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

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

A257146 Primitive non-solvable numbers: elements of A056866 not divisible by any earlier term.

Original entry on oeis.org

60, 168, 1092, 2448, 5616, 6072, 25308, 29120, 32736, 39732, 51888, 74412, 150348, 194472, 285852, 546312, 612468, 1285608, 1934868, 2097024, 2165292, 2588772, 3594432, 3822588, 5848428, 6324552, 7174332, 8487168, 9095592, 10626828, 11332452, 12576732, 14467068, 15331992, 15927348

Offset: 1

Charles R Greathouse IV, Apr 16 2015

nonn

A number is solvable if and only if it is a positive multiple of a member of this sequence.

There is 1 member of this sequence up to 10^2, 2 up to 10^3, 6 up to 10^4, 12 up to 10^5, 17 up to 10^6, 29 up to 10^7, 49 up to 10^8, 89 up to 10^9, 169 up to 10^10, 321 up to 10^11, 616 up to 10^12, 1188 up to 10^13, 2351 up to 10^14, 4679 up to 10^15, 9350 up to 10^16, 18866 up to 10^17, 38157 up to 10^18, 77534 up to 10^19, 158048 up to 10^20, 323358 up to 10^21, 663159 up to 10^22, and 1363304 up to 10^23. - Charles R Greathouse IV, Sep 16 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A056866.

list(lim)={
  my(v=List(),t);
  forprime(p=2,,
    t=(4^p-1)<lim,break);
    listput(v,t)
  );
  forprime(p=3,,
    t=3^p*(9^p-1)/2;
    if(t>lim,break);
    listput(v,t)
  );
  forprime(p=7,,
    t=p*(p^2-1)/2;
    if(t>lim,break);
    listput(v,t)
  );
  forprime(p=3,,
    t=4^p*(4^p+1)*(2^p-1);
    if(t>lim,break);
    listput(v,t)
  );
  if(lim>=5616, listput(v, 5616));
  v=Set(v);
  for(i=1,#v,
    if(v[i]==60 && i>1, next); \\ see below
    for(j=i+1,#v,
      if(v[j]%v[i]==0, v[j]=60) \\ delete values by setting to v[1]
    )
  );
  Set(v); \\ remove duplicates to combine all 60s
}

a(n) >> n^3 log^3 n. - Charles R Greathouse IV, Apr 20 2015

A131933 a(n) = A056866(n)/4.

Original entry on oeis.org

15, 30, 42, 45, 60, 75, 84, 90, 105, 120, 126, 135, 150, 165, 168, 180, 195, 210, 225, 240, 252, 255, 270, 273, 285, 294, 300, 315, 330, 336, 345, 360, 375, 378, 390, 405, 420, 435, 450, 462, 465, 480, 495, 504, 510, 525, 540, 546, 555, 570, 585, 588, 600, 612

Offset: 1

Artur Jasinski, Jul 30 2007

nonn

All orders of nonsolvable groups A056866 are divisible by 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001034, A060793, A056866, A056868, A131932.

A001034 Orders of noncyclic simple groups (without 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, 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).

A056868 Numbers that are not nilpotent numbers.

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, 75, 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, 120

Offset: 1

N. J. A. Sloane, Sep 02 2000

A number is nilpotent if every group of order n is nilpotent.

The sequence "Numbers of the form (k*i + 1)*k*j with i, j >= 1 and k >= 2" agrees with this for the first 146 terms but then differs. Cf. A300737. - Gionata Neri, Mar 11 2018

			From _Bernard Schott_, Dec 19 2021: (Start)
There are 2 groups with order 6: C_6 that is cyclic so nilpotent, and the symmetric group S_3 that is not nilpotent, hence 6 is a term.
There are also 2 groups with order 10: C_10 that is cyclic so nilpotent, and the dihedral group D_10 that is not nilpotent, hence 10 is another term. (End)

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, pp. 631-634.

Complement of A056867.
Subsequence of A060652; A068919 is a subsequence.
Cf. A003277, A051532, A056866, A027748, A124010, A300737.

a056868 n = a056868_list !! (n-1)
a056868_list = filter (any (== 1) . pks) [1..] where
   pks x = [p ^ k `mod` q | let fs = a027748_row x, q <- fs,
                            (p,e) <- zip fs $ a124010_row x, k <- [1..e]]
-- Reinhard Zumkeller, Jun 28 2013

nilpotentQ[n_] := With[{f = FactorInteger[n]}, Sum[ Boole[ Mod[p[[1]]^p[[2]], q[[1]]] == 1], {p, f}, {q, f}]] == 0; Select[ Range[120], !nilpotentQ[#]& ] (* Jean-François Alcover, Sep 03 2012 *)

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(1)))); 0 \\ Charles R Greathouse IV, Sep 18 2012

n is in this sequence if p^k = 1 mod q for primes p and q dividing n such that p^k divides n. - Charles R Greathouse IV, Aug 27 2012

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

A060793 Orders of finite perfect groups (groups such that G = G' where G' is the commutator subgroup of G).

Original entry on oeis.org

1, 60, 120, 168, 336, 360, 504, 660, 720, 960, 1080, 1092, 1320, 1344, 1920, 2160, 2184, 2448, 2520, 2688, 3000, 3420, 3600, 3840, 4080, 4860, 4896, 5040, 5376, 5616, 5760, 6048, 6072, 6840, 7200, 7500, 7560, 7680, 7800, 7920, 9720, 9828, 10080, 10752

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 26 2001

nonn

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]

Since a non-cyclic simple group is perfect this sequence contains A001034 and since a perfect group is non-solvable this sequence is a subsequence of A056866 (apart from the initial term).

			A_{5} is perfect since it is equivalent to A_{5}'.

D. Holt and W. Plesken, Perfect Groups, Oxford University Press, 1989.

Eric M. Schmidt, Table of n, a(n) for n = 1..300
Walter Feit, J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.
Index entries for sequences related to groups

Cf. A001034, A056866.

SizesPerfectGroups(); # Eric M. Schmidt, Nov 14 2013

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

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).

A064487 Order of twisted Suzuki group Sz(2^(2n + 1)), also known as the group 2B2(2^(2n + 1)).

Original entry on oeis.org

20, 29120, 32537600, 34093383680, 35115786567680, 36011213418659840, 36888985097480437760, 37777778976635853209600, 38685331082014736871587840, 39614005699412557795646504960, 40564799864499450381466515537920

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 15 2001

Every term in A056866 is divisible by 12 or 20. Those terms that are not divisible by 12 are divisible by a term from this sequence. - Charles R Greathouse IV via Danny Rorabaugh, Apr 21 2015

For n >= 3, a(n) has at least 5 distinct prime factors. See my link for a proof. - Jianing Song, Apr 04 2022

R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14.
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].

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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], p. xvi. See ATLAS v. 3
Jianing Song, Proof that a(n) has at least 5 distinct prime factors for n >= 3
Michio Suzuki, A new type of simple groups of finite order, Proc Natl Acad Sci U S A. 46:6 (1960), pp. 868-870.
Index entries for linear recurrences with constant coefficients, signature (1360,-365568,22282240,-268435456).

Cf. A037250, A064583. A257391 is a subsequence.

GAP
```
g := Sz(32); s := Size(g);
```

[ #Sz(2^(2*n+1)) : n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

LinearRecurrence[{1360,-365568,22282240,-268435456},{20,29120,32537600,34093383680},20] (* Harvey P. Dale, Sep 08 2018 *)

a(n)=my(t=2^(2*n+1)); t^2*(t-1)*(t^2+1) \\ Charles R Greathouse IV, Apr 21 2015

Vec(20*(1+128*x)*(1-32*x+16384*x^2)/((1-16*x)*(1-64*x)*(1-256*x)*(1-1024*x)) + O(x^20)) \\ Colin Barker, Dec 25 2015

a(n) = q^4*(q^2-1)*(q^4+1), where q^2 = 2^(2*n+1).

G.f.: 20*(1+128*x)*(1-32*x+16384*x^2) / ((1-16*x)*(1-64*x)*(1-256*x)*(1-1024*x)). - Colin Barker, Dec 25 2015

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.

Original entry on oeis.org

29120, 32537600, 34093383680, 36011213418659840, 36888985097480437760, 38685331082014736871587840, 39614005699412557795646504960, 41538369916519054182462860998737920, 44601490313984496701256699111250939955118080, 45671926145323068271210017365594287580527984640

Offset: 1

Danny Rorabaugh, Apr 21 2015

5 divides (4^m+1) for odd m, so every term in this sequence is a multiple of 5 (A008587).
A064487(k) = 4^(2k+1)*(4^(2k+1)+1)*(2^(2k+1)-1), so this sequence is a subsequence of A064487.
Every non-solvable number (A056866) is divisible by 12 or 20. All non-solvable numbers not divisible by 12 (A008594) are divisible by a member of this sequence. In particular, every primitive non-solvable number (A257146) not divisible by 12 is in this sequence.
All terms are divisible by 320 and have at least 4 distinct prime factors. - Jianing Song, Apr 04 2022

See A056866 and A064487 for references.

Danny Rorabaugh, Table of n, a(n) for n = 1..100

Subsequence of A008587, A008602, A056866, and A064487.

Table[4^p (4^p+1)(2^p-1),{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 17 2024 *)

a(n)=my(p=prime(n+1)); 4^p*(4^p+1)*(2^p-1) \\ Charles R Greathouse IV, Apr 21 2015

[4^nth_prime(n)*(4^nth_prime(n)+1)*(2^nth_prime(n)-1) for n in range(2,12)]

a(n) = 4^p*(4^p+1)*(2^p-1) where p = A065091(n) = A000040(n+1).

cp's OEIS Frontend

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

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A257146 Primitive non-solvable numbers: elements of A056866 not divisible by any earlier term.

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A131933 a(n) = A056866(n)/4.

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A001034 Orders of noncyclic simple groups (without repetition).

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A056868 Numbers that are not nilpotent numbers.

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A060793 Orders of finite perfect groups (groups such that G = G' where G' is the commutator subgroup of G).

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GAP

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

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

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A064487 Order of twisted Suzuki group Sz(2^(2n + 1)), also known as the group 2B2(2^(2n + 1)).

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.