A090751 -id:A090751 - cp's OEIS frontend

A069513 Characteristic function of the prime powers p^k, k >= 1.

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Benoit Cloitre, Apr 15 2002

Also, number of Galois fields of order n. - Charles R Greathouse IV, Mar 12 2008

Also, number of abelian indecomposable groups of order n. - Kevin Lamoreau, Mar 13 2023

Daniel Forgues, Table of n, a(n) for n = 1..100000.

Cf. A246655, A010055, A001221, A001222, A008683, A090751, A000688, A361414.
The partial sums of this sequence give A025528. - Daniel Forgues, Mar 02 2009
Cf. A014963, A003418. - Enrique Pérez Herrero, Jun 01 2011

a069513 1 = 0
a069513 n = a010055 n  -- Reinhard Zumkeller, Mar 19 2013

A069513 := proc(n)
    if n = 1 then
        0 ;
    elif A001221(n) > 1 then
        0;
    else
        1 ;
    end if ;
end proc:
seq(A069513(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

A069513[n_]:=Boole[PrimeNu[n]==1]; A069513/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)

PARI
```
for(n=1,120,print1(omega(n)==1,","))
```

from sympy import primefactors
def A069513(n): return int(len(primefactors(n)) == 1) # Chai Wah Wu, Mar 31 2023

If n >= 2, a(n) = A010055(n).
a(n) = Sum_{d|n} bigomega(d)*mu(n/d); equivalently, Sum_{d|n} a(d) = bigomega(n); equivalently, Möbius transform of bigomega(n).
Dirichlet g.f.: ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s - 1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = floor(1/A001221(n)), for n > 1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = - Sum_{d|n} mu(d)*bigomega(d), where bigomega = A001222. - Ridouane Oudra, Oct 29 2024
a(n) = - Sum_{d|n} mu(d)*omega(d), where omega = A001221. - Ridouane Oudra, Jul 30 2025

Moved original definition to formula line. Used comment (that I previously added) as definition. - Daniel Forgues, Mar 08 2009

Edited by Franklin T. Adams-Watters, Nov 02 2009

A338757 Number of splitting-simple groups of order n; number of nontrivial groups of order n that are not semidirect products of proper subgroups.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 19, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Jianing Song, Nov 07 2020

The other names for groups of this kind include "semidirectly indecomposable groups" or "inseparable groups". Note that the following are equivalent definitions for a nontrivial group to be a splitting-simple group:
- It is not the (internal) semidirect product of proper subgroups;
- It is not isomorphic to the (external) semidirect product of nontrivial groups;
- It has no proper nontrivial normal subgroups with a permutable complement.
- It is the non-split extension of every proper nontrivial normal subgroup by the corresponding quotient group.
Also note that being simple is a stronger condition than being splitting-simple, while being directly indecomposable (see A090751) is weaker.
a(p^e) >= 1 since C_p^e cannot be written as the semidirect product of proper subgroups. For e >= 3, a(2^e) >= 2 by the existence of the generalized quaternion group of order 2^e, which is the only non-split extension of C_2^(e-1) by C_2 other than C_2^e.
The smallest numbers here with a(n) > 0 that are not prime powers are 48, 60, 120, 144, 168, 192, 240, 320, 336, 360 and so on. Are there any odd numbers n that are not prime powers satisfying a(n) > 0 ?
Conjecture: a(n) = 0 for squarefree n which is not a prime.
The conjecture that a(n) = 0 for nonprime squarefree n is true. Proof: It is known that every group G of squarefree order is supersolvable; hence G contains a normal series with prime cyclic factors. Since every Sylow subgroup of G is prime cyclic, these cyclic factors are isomorphic to the Sylow subgroups of G. Let P be one such factor; then for an appropriate M in G, P = G/M, where |G| = |P|*|M|. By the Schur-Zassenhaus theorem, G is a semidirect product of M and P, and a(n) = 0 when n is squarefree. - Miles Englezou, Oct 24 2024

			a(48) = 1 because the binary octahedral group, which is of order 48, cannot be written as the semidirect product of proper subgroups.
a(16) = 2, and the corresponding groups are C_16 and Q_16 (generalized quaternion group of order 16).
a(81) = 2, and the corresponding groups are C_81 and SmallGroup(81,10).
a(64) = 19, and the corresponding groups are SmallGroup(64,i) for i = 1, 11, 13, 14, 19, 22, 37, 43, 45, 49, 54, 79, 81, 82, 160, 168, 172, 180 and 245.
For n = 60 or 168, the unique simple group is the only group of order n that cannot be written as the semidirect product of proper subgroups, hence a(60) = a(168) = 1. [The unique simple groups are respectively Alt(5) and PSL(2,7). - _Bernard Schott_, Nov 08 2020]
For n = 12, we have C_12 = C_3 X C_4, C_6 X C_2 = C_6 X C_2, D_6 = C_6 : C_2, Dic_12 = C_3 : C_4 and A_4 = (C_2 X C_2) : C_3, all of which can be written as the semidirect product of nontrivial groups. So a(12) = 0.

Jianing Song, Table of n, a(n) for n = 1..255
Tim Dokchitser, Group extensions
The Group Properties Wiki, Splitting-simple group
The Group Properties Wiki, Permutable complements
Jianing Song, List of splitting-simple groups of order up to 255 by ID in GAP's SmallGroup library
Index entries for sequences related to groups

Cf. A000001, A090751 (number of directly indecomposable groups of order n), A001034, A120944.

IsSplittingSimple := function(G)
  local c, l, i;
  c := NormalSubgroups(G);
  l := Length(c);
  if l > 1 then
    for i in [2..l-1] do
    if Length(ComplementClassesRepresentatives(G,c[i])) > 0 then
      return false;
    fi;
    od;
    return true;
  else
    return false;
  fi;
end;
A338757 := n -> Length(AllSmallGroups( n, IsSplittingSimple ));

For primes p != q:
a(p) = a(p^2) = 1; a(p^3) = 2 for p = 2, 1 otherwise;
a(p^4) = 2 for p = 2 or 3, 1 otherwise;
a(pq) = 0;
a(4p) = a(8p) = 0, p > 2.
a(n) <= A090751(n) for all n, and the equality holds if n = 1, p, p^2 for primes p or n = pq for primes p < q and p does not divide q-1.
a(A001034(k)) >= 1, since A001034 lists the orders of (non-Abelian) simple groups.
a(A120944(n)) = 0. - Miles Englezou, Oct 24 2024

A361414 Number of non-abelian indecomposable groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 7, 0, 2, 0, 2, 1, 1, 0, 6, 0, 1, 2, 1, 0, 1, 0, 33, 0, 1, 0, 4, 0, 1, 1, 5, 0, 2, 0, 1, 0, 1, 0, 23, 0, 2, 0, 2, 0, 6, 1, 5, 1, 1, 0, 3, 0, 1, 1, 200, 0, 1, 0, 2, 0, 1, 0, 19, 0, 1, 1, 1, 0, 2, 0, 24, 8, 1, 0, 3, 0

Offset: 1

Kevin Lamoreau, Mar 10 2023

nonn

			a(16) = 7 because 1 of the 8 indecomposable groups of order 16 is abelian and 2 of the 9 non-abelian groups of order 16 are decomposable, leaving 7 non-abelian indecomposable groups of order 16.

Kevin Lamoreau, Table of n, a(n) for n = 1..2047

Cf. A090751, A060689, A069513.

a(n) = A090751(n) - A069513(n).

A094448 Number of indecomposable groups of order 2^n.

Original entry on oeis.org

0, 1, 1, 3, 8, 34, 201, 2000, 53410, 10435175, 49476809194

Offset: 0

Paul Boddington, Jun 04 2004

See A090751 for definition of indecomposable.

Cf. A000679.

Inverse Euler transform of A000679. - Franklin T. Adams-Watters, Sep 22 2006

More terms from Franklin T. Adams-Watters, Sep 22 2006

a(0) changed to 0 by Eric M. Schmidt, Jun 07 2014

A109230 Number of indecomposable groups with n conjugacy classes.

Original entry on oeis.org

0, 1, 2, 3, 8, 6, 12, 14

Offset: 1

Paul Boddington, Aug 19 2005

See A090751 for the definition of indecomposable. The comments there imply that A073043 has Dirichlet generating function prod((1-n^(-s))^(-a(n)),n>=2).

A122697 Number of indecomposable partitions of n.

Original entry on oeis.org

0, 2, 3, 2, 7, 5, 15, 14, 24, 28, 56, 52, 101, 105, 155, 189, 297, 310, 490, 536, 747, 890, 1255, 1380, 1930, 2234, 2928, 3433, 4565, 5133, 6842, 7881, 9975, 11716, 14778, 17006, 21637, 25035, 30882, 35972, 44583, 51200, 63261, 73115, 88459, 103048

Offset: 1

Franklin T. Adams-Watters, Sep 22 2006

nonn

A partition is indecomposable if it is not [1] and cannot be represented as the product of two smaller partitions, where the product of two partitions is the multiset of all products of parts from the two multiplicands. Another way to define the product of partitions is to regard the partition as a finite sequence b(k) being the number of parts of size k; then the Dirichlet g.f. of b * c is the product of the Dirichlet g.f.s of b and c.

			The product of [2,2,1] * [2,1,1] is the partition with parts:
4 4 2
2 2 1
2 2 1
which is [4^2,2^5,1^2]. In terms of Dirichlet g.f.s, this is (2*2^s + 1^s) * (2^s + 2*1^s) = (2*4^s + 5*2^s + 2*1^s).
Of the partitions of 6, [6] = [3] * [2], [4,2] = [2] * [2,1], [3^2] = [3] * [1^2], [2^3] = [2] * [1^3], [2^2,1^2] = [2,1] * [1^2] and [1^6] = [1^3] * [1^2]. This leaves [5,1], [4,1^2], [3,2,1], [3,1^3] and [2,1^4] as the 5 indecomposable partitions of 6.

Cf. A000041, A090751.

The (formal) Dirichlet generating function for A000041 is Product_{n>1} 1/(1-n^{-s})^a(n). (Formal because this g.f. does not converge for any value of s.)

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A069513 Characteristic function of the prime powers p^k, k >= 1.

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A338757 Number of splitting-simple groups of order n; number of nontrivial groups of order n that are not semidirect products of proper subgroups.

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A361414 Number of non-abelian indecomposable groups of order n.

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A094448 Number of indecomposable groups of order 2^n.

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A109230 Number of indecomposable groups with n conjugacy classes.

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A122697 Number of indecomposable partitions of n.

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A243592 Numbers n such that there is no indecomposable group of order n.

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