A060689 -id:A060689 - cp's OEIS frontend

A239699 Numbers n such that the number of Abelian groups of order n is equal to the number of non-Abelian groups of order n.

6, 10, 14, 21, 22, 26, 28, 34, 38, 39, 44, 46, 55, 57, 58, 62, 63, 74, 76, 82, 86, 92, 93, 94, 105, 106, 111, 117, 118, 122, 124, 129, 134, 142, 146, 155, 158, 165, 166, 172, 178, 183, 188, 194, 195, 201, 202, 203, 205, 206, 214, 218, 219, 226, 231, 236, 237

Offset: 1

Michel Lagneau, Mar 24 2014

nonn

Numbers n such that A000688(n) = A060689(n).

			6 is in the sequence because there are 2 groups of order 6: 1 commutative group and 1 non-commutative group. Then A000688(6) = A060689(6) = 1.
44 is in the sequence because there are 4 groups of order 44: 2 commutative groups and 2 non-commutative groups. Then A000688(44) = A060689(44) = 2.

Michel Lagneau, Table of n, a(n) for n = 1..390

Cf. A000688, A060689, A238877.

lst:={};f[n_]:=Times@@PartitionsP/@Last/@FactorInteger@n;g[n_]:=FiniteGroupCount[n]-FiniteAbelianGroupCount[n];Do[If[f[n]==g[n],AppendTo[lst,n]],{n,500}];lst

A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 3, 0, 0, 1, 0, 0, 1, 1, 0, 5, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 1, 0, 5, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 3, 0, 1, 1, 0, 0, 5, 1, 0, 5, 0, 1, 0, 1, 0, 0, 1, 3, 0, 0, 1, 0, 3, 0, 0

Offset: 1

Chai Wah Wu, Aug 17 2024

nonn

O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211-219 (1895).

Chai Wah Wu, Table of n, a(n) for n = 1..10000
I. Ganev, Groups of a Square-Free order, Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 7 (2010).

Cf. A000001, A000688, A005117, A060689, A375491.

from math import isqrt, prod
from itertools import combinations
from sympy import mobius, primefactors, npartitions, factorint
def A375483(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    a = set(primefactors(m))
    return sum(prod((p**sum(1 for q in b if q%p==1)-1)//(p-1) for p in a-set(b)) for l in range(0,len(a)+1) for b in combinations(a,l))-prod(map(npartitions, factorint(m).values()))

a(n) = A060689(A005117(n)) = A375491(n) - A000688(A005117(n)).

A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

Original entry on oeis.org

1, 6, 12, 16, 24, 32, 48, 64, 96, 128, 256, 512, 1024, 2048

Offset: 1

Ben Branman, Feb 29 2012

			For a(n)=12, there are 2 Abelian groups and 3 nonabelian groups, so the ratio A000001(12)/A000688(12)=5/2=2.5, which beats the previous record of 2, so 12 is in the sequence.

H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
H. U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
H. U. Besche and B. Eick, The Groups of Order at Most 1000 Except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
Eric Weisstein's World of Mathematics, Finite Group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Index entries for sequences related to groups

Cf. A000001, A000688, A060689, A051532.

s = {1}; a = 1; Do[b = FiniteGroupCount[n]/FiniteAbelianGroupCount[n];
  If[b > a, a = b; AppendTo[s, n]], {n, 1, 2047}]; s

a(14) from Eric M. Schmidt, Aug 02 2012

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A239699 Numbers n such that the number of Abelian groups of order n is equal to the number of non-Abelian groups of order n.

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A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.

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A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

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