A034382 -id:A034382 - cp's OEIS frontend

A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

N. J. A. Sloane

Equivalently, number of Abelian groups with n conjugacy classes. - Michael Somos, Aug 10 2010
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^(k+1) = x. - Franklin T. Adams-Watters, Oct 20 2006
Range is A033637.
a(n) = 1 if and only if n is from A005117 (squarefree numbers). See the Ahmed Fares comment there, and the formula for n>=2 below. - Wolfdieter Lang, Sep 09 2012
Also, from a theorem of Molnár (see [Molnár]), the number of (non-isomorphic) abelian groups of order 2*n + 1 is equal to the number of non-congruent lattice Z-tilings of R^n by crosses, where a "cross" is a unit cube in R^n for which at each facet is attached another unit cube (Z, R are the integers and reals, respectively). (Cf. [Horak].) - L. Edson Jeffery, Nov 29 2012
Zeta(k*s) is the Dirichlet generating function of the characteristic function of numbers which are k-th powers (k=1 in A000012, k=2 in A010052, k=3 in A010057, see arXiv:1106.4038 Section 3.1). The infinite product over k (here) is the number of representations n=product_i (b_i)^(e_i) where all exponents e_i are distinct and >=1. Examples: a(n=4)=2: 4^1 = 2^2. a(n=8)=3: 8^1 = 2^1*2^2 = 2^3. a(n=9)=2: 9^1 = 3^2. a(n=12)=2: 12^1 = 3*2^2. a(n=16)=5: 16^1 = 2*2^3 = 4^2 = 2^2*4^1 = 2^4. If the e_i are the set {1,2} we get A046951, the number of representations as a product of a number and a square. - R. J. Mathar, Nov 05 2016
See A060689 for the number of non-abelian groups of order n. - M. F. Hasler, Oct 24 2017
Kendall & Rankin prove that the density of {n: a(n) = m} exists for each m. - Charles R Greathouse IV, Jul 14 2024

			a(1) = 1 since the trivial group {e} is the only group of order 1, and it is Abelian; alternatively, since the only factorization of 1 into prime powers is the empty product.
a(p) = 1 for any prime p, since the only factorization into prime powers is p = p^1, and (in view of Lagrange's theorem) there is only one group of prime order p; it is isomorphic to (Z/pZ,+) and thus Abelian.
From _Wolfdieter Lang_, Jul 22 2011: (Start)
a(8) = 3 because 8 = 2^3, hence a(8) = pa(3) = A000041(3) = 3 from the partitions (3), (2, 1) and (1, 1, 1), leading to the 3 factorizations of 8: 8, 4*2 and 2*2*2.
a(36) = 4 because 36 = 2^2*3^2, hence a(36) = pa(2)*pa(2) = 4 from the partitions (2) and (1, 1), leading to the 4 factorizations of 36: 2^2*3^2, 2^2*3^1*3^1, 2^1*2^1*3^2 and 2^1*2^1*3^1*3^1.
(End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
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. Speiser, Die Theorie der Gruppen von endlicher Ordnung, 4. Auflage, Birkhäuser, 1956.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tak-Shing T. Chan, and Y.-H. Yang, Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171.
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
Steven R. Finch, Abelian Group Enumeration Constants [Broken link]
Steven R. Finch, Abelian Group Enumeration Constants [broken link?] [From the Wayback machine]
P. Horak, Error-correcting codes and Minkowski's conjecture, Tatra Mt. Math. Publ., 45 (2010), p. 40.
B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
D. G. Kendall, R. A. Rankin, On the number of Abelian groups of a given order, Q. J. Math. 18 (1947) 197-208.
Nobushige Kurokawa and Masato Wakayama, Zeta extensions. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. MR1930216 (2003h:11112).
E. Molnár, Sui mosaici dello spazio di dimensione n, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.
H.-E. Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
Marko Riedel, Counting Abelian Groups, Mathematics Stack Exchange, October 2014.
Laszlo Toth, A note on the number of abelian groups of a given order, arXiv:1203.6473 [math.NT], (2012).
Eric Weisstein's World of Mathematics, Abelian Group
Eric Weisstein's World of Mathematics, Finite Group
Eric Weisstein's World of Mathematics, Kronecker Decomposition Theorem
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A021002, A060689, A000041, A000961, A001055, A005361, A034382, A046054, A046055, A046056, A046101, A050360, A055653, A057521, A101872 (bisection), A101876 (quadrisection), A124010, A050361, A051532, A129667 (Dirichlet inverse), A188585.

Cf. A080729 (Dgf at s=2), A369634 (Dgf at s=3).

a000688 = product . map a000041 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,ans): od: # James Sellers, Dec 07 2000

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v, Sep 22 2006 *)
Table[FiniteAbelianGroupCount[n], {n, 200}] (* Requires version 7.0 or later. - Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)

A000688(n)=local(f);f=factor(n);prod(i=1,matsize(f)[1],numbpart(f[i,2])) \\ Michael B. Porter, Feb 08 2010

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,numbpart(f[i])) \\ Charles R Greathouse IV, Apr 16 2015

from sympy import factorint, npartitions
from math import prod
def A000688(n): return prod(map(npartitions,factorint(n).values())) # Chai Wah Wu, Jan 14 2022

def a(n):
    F=factor(n)
    return prod([number_of_partitions(F[i][1]) for i in range(len(F))])
# Ralf Stephan, Jun 21 2014

Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.
a(2n) = A101872(n).
a(n) = Product_{j = 1..N(n)} A000041(e(j)), n >= 2, if
  n  = Product_{j = 1..N(n)} prime(j)^e(j), N(n) = A001221(n). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - Wolfdieter Lang, Jul 23 2011
In terms of the cycle index of the symmetric group: Product_{q=1..m} [z^{v_q}] Z(S_v) 1/(1-z) where v is the maximum exponent of any prime in the prime factorization of n, v_q are the exponents of the prime factors, and Z(S_v) is the cycle index of the symmetric group on v elements. - Marko Riedel, Oct 03 2014
Dirichlet g.f.: Sum_{n >= 1} a(n)/n^s = Product_{k >= 1} zeta(ks) [Kendall]. - Álvar Ibeas, Nov 05 2014
a(n)=2 for all n in A054753 and for all n in A085987.  a(n)=3 for all n in A030078 and for all n in A065036.  a(n)=4 for all n in A085986.  a(n)=5 for all n in A030514 and for all n in A178739.  a(n)=6 for all n in A143610. - R. J. Mathar, Nov 05 2016
A050360(n) = a(A025487(n)). a(n) = A050360(A101296(n)). - R. J. Mathar, May 26 2017
a(n) = A000001(n) - A060689(n). - M. F. Hasler, Oct 24 2017
From Amiram Eldar, Nov 01 2020: (Start)
a(n) = a(A057521(n)).
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A021002. (End)
a(n) = A005361(n) except when n is a term of A046101, since A000041(x) = x for x <= 3. - Miles Englezou, Feb 17 2024
Inverse Moebius transform of A188585: a(n) = Sum_{d|n} A188585(d). - Amiram Eldar, Jun 10 2025

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

Original entry on oeis.org

1, 2, 2, 3, 18, 6, 16, 180, 144, 36, 30, 2040, 3240, 1740, 380, 360, 43170, 81000, 70740, 31680, 6390, 840, 1400112, 2589510, 2976960, 2055480, 832230, 157962, 15360, 110488616, 117733728, 144285960, 130781280, 79626120, 30004128, 5396888, 68040, 30647444544, 9223088112, 8744866704, 8997002280, 7154708400, 4005012816, 1421659512, 243179064

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
    1;
    2,     2;
    3,    18,     6;
   16,   180,   144,    36;
   30,  2040,  3240,  1740,   380;
  360, 43170, 81000, 70740, 31680, 6390;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Index entries for sequences related to monoids

Row sums give A058155.
Column 1: A034382.
Main diagonal: A055512.
Cf. A058142 (isomorphism classes), A058157, A058160.

T(n, k) = A058160(n, k)*n.

Terms a(30) and beyond from Andrew Howroyd, Feb 15 2022

A058162 Number of labeled Abelian groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 4, 6, 60, 120, 1920, 7560, 90720, 362880, 13305600, 39916800, 1037836800, 10897286400, 265686220800, 1307674368000, 66691392768000, 355687428096000, 20274183401472000, 202741834014720000

Offset: 1

Christian G. Bower, Nov 15 2000, Mar 12 2008

nonn

The distinction here between labeled and unlabeled Abelian groups is analogous to the distinction between unlabeled rooted trees (A000081) and labeled rooted trees (A000169).
That is, the number of Cayley tables. - Artur Jasinski, Mar 12 2008
Number of Latin squares in dimension n with first row and first column 1,2,3 ..., n which are associative and commutative (Abelian). Each of these squares is isomorphic with the Cayley table of one of the existed Abelian group in dimension n. - Artur Jasinski, Nov 02 2005. Cf. A111341.

			The 2 unlabeled Abelian groups of order 4 are C4 and C2^2. The 4 labeled Abelian groups whose identity is "0" consist of 3 of type C4 (where the nongenerator can be "2", "3", or "4") and 1 of type C2^2.

Max Alekseyev, Table of n, a(n) for n = 1..100
C. J. Hillar, D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR]
Index entries for sequences related to groups

Cf. A000688, A058160, A058161, A058163.

a(n) = A034382(n) / n. Formula for A034382 is based on the fundamental theorem of finite Abelian groups and the formula given by Hillar and Rhea (2007).

a(16) and a(21) corrected by Max Alekseyev, Sep 12 2019

A034381 Number of labeled cyclic groups with n elements.

Original entry on oeis.org

1, 2, 3, 12, 30, 360, 840, 10080, 60480, 907200, 3991680, 119750400, 518918400, 14529715200, 163459296000, 2615348736000, 22230464256000, 1067062284288000, 6758061133824000, 304112751022080000, 4257578514309120000, 112400072777760768000, 1175091669949317120000

Offset: 1

Christian G. Bower

This sequence is strictly increasing, since a(n) = n!/phi(n) > n!/n = (n-1)! >= a(n-1) for n >= 2. - Jianing Song, Mar 02 2024

Jianing Song, Table of n, a(n) for n = 1..450
Index entries for sequences related to groups

Cf. A000010, A000142, A034382, A034383, A058161.

a(n) = n! / eulerphi(n) \\ Jianing Song, Mar 02 2024

a(n) = n!/phi(n).

a(n) = A000142(n)/A000010(n) = n*A058161(n).

a(21) onwards from Jianing Song, Mar 02 2024

A034383 Number of labeled groups.

Original entry on oeis.org

1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, 259459200, 518918400, 16605388800, 163459296000, 10353459916800, 22230464256000, 1867358997504000, 6758061133824000, 648773868847104000, 5474029518397440000, 122618261212102656000

Offset: 1

Christian G. Bower

nonn

From Jianing Song, Mar 02 2024: (Start)
In other words, number of ways to define a group structure on a set of n elements. Note that for a group G, a group structure on the set G is given by mapping (x,y) to sigma^(-1)(sigma(x)*sigma(y)), where sigma is a bijection on the set G; sigma and sigma' give the same structure if and only if sigma' is the composition of a group automorphism of G and sigma.
By definition, a(n) = A034381(n) if n in A003277, otherwise a(n) > A034381(n). The indices of records of a(n)/A034381(n) among the known terms are 1, 4, 8, 16, 24, 32, 48, 64, 96, 128, 192, with a(192)/A034381(192) = 122774329/1640520 ~ 74.8.
Also by definition, a(n) >= A000001(n)*n!/A059773(n). If the conjecture A059773(2^r) = A002884(r) is true, then A059773(2^r) <= 2^(r^2), while A000001(2^r) >= 2^((2/27)*r^2*(r-6)) (see the Math Stack Exchange link below), so a(2^r)/A034381(2^r) tends to infinity quickly as r tends to infinity.
The sequence is strictly increasing for the first 256 terms (a(256) > A034381(256) > A034381(255) = a(255) since 255 is in A003277). On the other hand, assuming that A059773(2^r) = A002884(r), then a(2^20)/(2^20)! >= A000001(2^20)/A002884(20) > 99798.4, while a(2^20+1)/(2^20)! = A034381(2^20+1)/(2^20)! = (2^20+1)/phi(2^20+1) since 2^20+1 = 17*61681 is in A003277, so we would have a(2^20) > a(2^20+1). It is conjectured a(2^r) > a(2^r+1) for all sufficiently large r. (End)

Stephen A. Silver, Table of n, a(n) for n = 1..255
H. U. Besche, The Small Groups library
Math Stack Exchange, Known bounds for the number of groups of a given order.
Index entries for sequences related to groups

Cf. A000001, A034381, A034382, A058157, A058163.

A034383 := function(n) local fn, sum, k; fn := Factorial(n); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn / Size(AutomorphismGroup(SmallGroup(n,k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013

a(n) = n * A058163(n).

a(n) = Sum n!/|Aut(G)|, where the sum is taken over the different products G of cyclic groups with |G| = n.

More terms from Stephen A. Silver, Feb 10 2013

A370360 Number of labeled semisimple rings with n elements.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 24409921536000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Paul Laubie, Mar 05 2024

nonn

Using the Artin-Wedderburn theorem, a finite semisimple ring is a product of matrix algebras over finite field. In particular, if n is squarefree then any semisimple ring of cardinal n is commutative. One can be more precise, indeed all semisimple rings with n elements are commutative if and only if the only 4th power that divides n is 1.
The analogous sequences for abelian groups and cyclic groups are A034382 and A034381, respectively.
In the case of commutative semisimple rings, we get the factorial numbers.

			For n=4, we have two possible rings: F_4 and F_2 X F_2. We use the notation F_q to denote the finite field with q elements. To compute a(4) we need to know how many ring automorphisms F_4 and F_2 X F_2 admit. For F_4, we have that Aut(F_4) is generated by the Frobenius morphism, hence we have 2 automorphisms. For F_2 X F_2, the only nontrivial automorphism is exchanging the two coordinates, hence we also have 2 automorphisms. Hence:
a(4) = 24/2 + 24/2 = 24.
We can compute a(2^k) for some small values of k:
a(4) = 4! = 24,
a(8) = 8!,
a(16) = 16! + 16!/6,
a(32) = 32! + 32!/6,
a(64) = 64! + 64!/12 + 64!/12,
a(128) = 128! + 128!/36 + 128!/18 + 128!/12,
...

Paul Laubie, A Github repository with a code to compute the terms of the form a(p^n).

Cf. A000142, A005117, A038538.

Cf. A034382, A034381.

If n is squarefree then we have a(n) = n!. More precisely, a(n) = n! if and only if the only 4th power that divides n is 1. In particular, n=16 is the smallest n such that a(n) is different from n!.

If n and m are relatively prime, then a(n*m) = (n*m)!*a(n)*a(m)/(n!*m!).

cp's OEIS Frontend

A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Sage

Formula

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A058162 Number of labeled Abelian groups with a fixed identity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A034381 Number of labeled cyclic groups with n elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A034383 Number of labeled groups.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Formula

Extensions

A370360 Number of labeled semisimple rings with n elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula