A021002 -id:A021002 - 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

A063966 Number of Abelian groups of order <= n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, 26, 28, 29, 31, 32, 33, 34, 37, 39, 40, 43, 45, 46, 47, 48, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 76, 77, 82, 84, 86, 87, 89, 90, 93, 94, 97, 98, 99, 100, 102, 103, 104, 106, 117, 118, 119, 120, 122

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Sep 04 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Hong-Quan Liu, On the number of abelian groups of a given order (supplement), Acta Arithmetica, Vol. 64, No. 3 (1993), pp. 285-296.
Eric Weisstein's World of Mathematics, Abelian Group.

Partial sums of A000688.
Cf. A063756.
Cf. A021002, A084892, A084893.

with(combinat): readlib(ifactors): total := 0: for n from 1 to 100 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,total+ans): total := total+ans: od:

Accumulate[Table[FiniteAbelianGroupCount[n], {n, 1, 200}]] (* Geoffrey Critzer, Dec 28 2014 *)

a(n) ~ c * n, where c = A021002 = Product_{k>=2} zeta(k). - Vaclav Kotesovec, Oct 26 2019

More accurately, a(n) = A021002 * n + A084892 * n^(1/2) + A084893 * n^(1/3) + O(n^(50/199 + eps)), where eps>0 is arbitrarily small (Liu, 1993). - Amiram Eldar, Sep 23 2023

More terms from James Sellers, Sep 26 2001

A085846 Decimal expansion of root of x = (1+1/x)^x.

Original entry on oeis.org

2, 2, 9, 3, 1, 6, 6, 2, 8, 7, 4, 1, 1, 8, 6, 1, 0, 3, 1, 5, 0, 8, 0, 2, 8, 2, 9, 1, 2, 5, 0, 8, 0, 5, 8, 6, 4, 3, 7, 2, 2, 5, 7, 2, 9, 0, 3, 2, 7, 1, 2, 1, 2, 4, 8, 5, 3, 7, 7, 1, 0, 3, 9, 6, 1, 6, 8, 5, 0, 6, 4, 8, 8, 0, 0, 9, 1, 5, 7, 7, 4, 3, 6, 2, 9, 0, 4, 2, 0, 1, 3, 8, 0, 4, 8, 2, 8, 2, 5, 6, 6, 1

Offset: 1

Eric W. Weisstein, Jul 05 2003

Equivalently, the root of x^(x+1) = (x+1)^x.
Also a root of 1/(x^(1/x)-1) - x = 0 and 1/(x^(1/x)-1/x-1) - x = 0, which also contains the root 5.50798565277317825758902... 1/(x^(1/x)-1) ~ Pi(x) and 1/(x^(1/x)-1/x-1) ~ Pi(x), which is a much better approximation. These roots also can be computed by the recurrences x = 1/(x^(1/x)-1) and x = 1/(x^(1/x)-1/x-1). - Cino Hilliard, Sep 13 2008
This constant is transcendental (Lord, 2002). - Amiram Eldar, Oct 29 2022

			2.2931662874118610315080282912508058643722572903271212485377103961...

Nicolae Anghel, Foias Numbers, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa, 2018) 26(3), 21-28.
Nick Lord, Two Other Transcendental Numbers Obtained by (Mis)calculating e, The Mathematical Gazette, Vol. 86, No. 505 (2002), pp. 103-105.
Eric Weisstein's World of Mathematics, Foias Constant.
Index entries for transcendental numbers.

Cf. A021002, A124930, A169862.

RealDigits[ FindRoot[x^(1/x) - (x + 1)^(1/(x + 1)) == 0, {x, 2}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v *)

solve(x=2,3,(1+1/x)^x-x) \\ Charles R Greathouse IV, Apr 14 2014

x satisfies x^(1/x) = (x+1)^(1/(x+1)). - Marco Matosic, Nov 25 2005

A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 7, 4, 4, 2, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 14, 4, 4, 7, 8, 2, 8, 2, 19, 4, 4, 4, 16, 2, 4, 4, 14, 2, 8, 2, 8, 8, 4, 2, 24, 4, 8, 4, 8, 2, 14, 4, 14, 4, 4, 2, 16, 2, 4, 8, 30, 4, 8, 2, 8, 4, 8, 2, 28, 2, 4, 8, 8, 4, 8, 2, 24, 12, 4, 2, 16, 4, 4, 4, 14, 2, 16

Offset: 1

Marc Bogaerts, Apr 04 2011

			For n=8; the divisors of 8 are 1,2,4,8. There are 1,1,2,3 abelian groups of these orders respectively, so a(n) = 1+1+2+3 = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000688, A000070.

trf:=function ( f, x )  # the Dirichlet convolution 1 * f
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i );
        end );
end;
nra:=function ( x )     # the number of Abelian Groups of order(n)
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return trf( nra, n );
end;

with(combinat): with(numtheory):
a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2011

InverseMobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}]; Do[b[[i]] = Plus @@ a[[Divisors[i]]], {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; InverseMobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 07 2011 *)
f[0] = 1; f[e_] := f[e] = f[e - 1] + PartitionsP[e]; a[1] = 1; a[n_] := Times @@ (f[Last[#]] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

A000688(n)={local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))}
A188581(n)=sumdiv(n,d,A000688(d))
r=vector(66,n,A188581(n)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */

a(n) = Sum_{d | n} A000688(d).
Multiplicative with a(p^e) = A000070(e). - Amiram Eldar, Sep 09 2020
Dirichlet g.f.: zeta(s)^2 * Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ n*((log(n) + 2*gamma - 1)*f(1) + f'(1)), where f(1) = Product_{k>=2} zeta(k) = A021002 = 2.1955691982567064617939..., f'(1) = f(1) * Sum_{k>=2} k*zeta'(k)/zeta(k) = -5.0385164470942955610707128990779476296197... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 21 2021

A084892 Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).

Original entry on oeis.org

1, 4, 6, 4, 7, 5, 6, 6, 3, 0, 1, 6, 3, 8, 3, 1, 1, 3, 1, 6, 9, 9, 9, 7, 6, 0, 9, 1, 2, 2, 0, 4, 2, 1, 9, 2, 6, 3, 8, 1, 1, 7, 3, 0, 3, 4, 7, 9, 6, 9, 6, 0, 2, 5, 1, 6, 9, 2, 6, 9, 3, 9, 7, 5, 2, 0, 1, 2, 7, 5, 7, 9, 1, 0, 4, 4, 9, 2, 6, 3, 5, 2, 5, 2, 9, 1, 8, 1, 7, 4, 2, 3, 5, 1, 0, 2, 2, 7, 0, 9, 4, 1

Offset: 2

Eric W. Weisstein, Jun 10 2003

This constant, A_2, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_3 = A084893. - Amiram Eldar, Oct 16 2020

			-14.64756630163831131699976...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000688, A021002, A063966, A084893.

m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/2]*Product[Zeta[j/2], {j, 3, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)

prodinf(k=1, if (k!=2, zeta(k/2), 1)) \\ Michel Marcus, Oct 16 2020

A084893 Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).

Original entry on oeis.org

1, 1, 8, 6, 9, 2, 4, 6, 1, 9, 7, 2, 7, 6, 4, 2, 8, 4, 6, 2, 6, 1, 6, 6, 9, 9, 3, 8, 1, 3, 7, 1, 1, 8, 0, 4, 8, 7, 8, 4, 8, 1, 4, 7, 7, 7, 0, 0, 0, 3, 6, 5, 8, 1, 3, 8, 9, 3, 3, 7, 7, 0, 9, 6, 8, 6, 7, 0, 8, 1, 5, 0, 4, 4, 2, 7, 8, 9, 8, 5, 9, 2, 1, 6, 1, 1, 1, 9, 0, 1, 8, 3, 4, 1, 2, 8, 9, 5, 3, 9, 5, 7

Offset: 3

Eric W. Weisstein, Jun 10 2003

This constant, A_3, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_2 = A084892. - Amiram Eldar, Oct 16 2020

			118.6924619727642846261669938137118...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000688, A021002, A063966, A084892.

m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/3]*Zeta[2/3]*Product[Zeta[j/3], {j, 4, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)

prodinf(k=1, if (k!=3, zeta(k/3), 1)) \\ Michel Marcus, Oct 16 2020

A104404 Number of groups of order n all of whose subgroups are normal.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 2, 12, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 5, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1

Offset: 1

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

A finite non-Abelian group has all of its subgroups normal precisely when it is the direct product of the quaternion group of order 8, a (possibly trivial) elementary Abelian 2-group, and an Abelian group of odd order. [Carmichael, p. 114] - Eric M. Schmidt, Jan 12 2014

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Hans Havermann, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Abelian Group.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Cf. A000001, A000688, A021002, A048651, A104488.

orders[n_]:=Map[Last, FactorInteger[n]]; b[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=b[e[n]]; h[n_]:=0; a[n_]:= b[n]+h[n];

a(n)={my(e=valuation(n, 2)); my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i]))*(numbpart(e) + (e>=3))} \\ Andrew Howroyd, Aug 08 2018

The number a(n) of all groups of order n all of whose subgroups are normal is given as a(n) = b(n) + h(n), where b(n) denotes the number of Abelian groups of order n and h(n) denotes the number of Hamiltonian groups of order n.
a(n) = A000688(n) + A104488(n). - Andrew Howroyd, Aug 08 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Sep 23 2023

Keyword:mult added by Andrew Howroyd, Aug 08 2018

A080729 Decimal expansion of the infinite product of zeta functions for even arguments.

Original entry on oeis.org

1, 8, 2, 1, 0, 1, 7, 4, 5, 1, 4, 9, 9, 2, 9, 2, 3, 9, 0, 4, 0, 6, 7, 2, 5, 1, 3, 2, 2, 2, 6, 0, 0, 6, 8, 4, 8, 5, 7, 8, 2, 6, 8, 0, 2, 8, 6, 4, 8, 2, 7, 1, 7, 5, 5, 0, 0, 2, 0, 9, 3, 8, 0, 0, 2, 8, 6, 0, 6, 5, 8, 8, 6, 7, 7, 0, 5, 4, 8, 8, 9, 3, 6, 3, 9, 6, 0, 2, 4, 9, 7, 5, 2, 1, 4, 5, 2, 9, 7, 6, 6, 1, 0, 9, 9

Offset: 1

Deepak R. N (deepak_rn(AT)safe-mail.net), Mar 08 2003

By elementary estimates, the constant lies in the open interval (Pi/6, exp(3/4)). - Bernd C. Kellner, May 18 2024

			1.82101745149929239040672513222600684857...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 658.
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Eric Weisstein's World of Mathematics, Abelian group.

Cf. A000688, A021002, A076813, A080730, A369634.

RealDigits[Product[Zeta[2n],{n,500}],10,110][[1]] (* Harvey P. Dale, Jan 31 2012 *)

prodinf(k=1, zeta(2*k)) \\ Vaclav Kotesovec, Jan 29 2024

Decimal expansion of zeta(2)*zeta(4)*...*zeta(2k)*...
If u(k) denotes the number of Abelian groups with group order k (A000688), then Product_{k>=1} zeta(2*k) = Sum_{k>=1} u(k)/k^2. - Benoit Cloitre, Jun 25 2003
Equals A021002/A080730. - Amiram Eldar, Jan 31 2024
This constant C is connected with the product of values of the Dedekind eta function on the upper imaginary axis. The product runs over the primes, where i is the imaginary unit: 1/C = Product_{prime p} (p^(1/12) * eta(i * log(p) / Pi)). - Bernd C. Kellner, May 18 2024

More terms from Benoit Cloitre, Mar 08 2003

A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...

Original entry on oeis.org

7, 5, 2, 0, 1, 0, 7, 4, 2, 3, 7, 7, 0, 2, 9, 1, 6, 1, 5, 2, 0, 6, 3, 6, 0, 7, 7, 4, 5, 5, 4, 3, 2, 5, 7, 6, 5, 6, 0, 7, 1, 8, 1, 4, 6, 9, 5, 9, 1, 2, 8, 5, 2, 6, 6, 9, 6, 3, 9, 9, 7, 9, 8, 3, 2, 6, 7, 2, 3, 5, 0, 5, 6, 8, 4, 6, 4, 7, 9, 7, 3, 7, 8, 6, 3, 9, 4, 7, 3, 6, 3, 7, 8, 0, 8, 6, 5, 4, 3, 7, 1, 0, 1, 3, 6

Offset: 0

Eric W. Weisstein, Jun 11 2003

			0.7520107423...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See p. 16.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1; arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000041, A000688, A021002, A084892, A084893, A272339.

digits = 10; m0 (* initial number of primes *) = 10^6; dm = 2*10^5; PP = PartitionsP; DP[n_] := DP[n] = (1/PP[n - 1] - 1 /PP[n]) // N[#, digits + 5]&; pmax = Prime[1000];
nmax[p_ /; p <= pmax] := nmax[p] = Module[{n}, For[n = 2, n < 1000, n++, If[Abs[1/PP[n - 1] - 1 /PP[n]]/p^n < 10^-100, Return[n]]]]; nmax[p_ /; p > pmax] := nmax[pmax];
s[p_] := Sum[DP[n]/p^n, {n, 2, nmax[p]}] ;
f[m_] := f[m] = Product[1 - s[p], {p, Prime[Range[m]]}]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2][[1]] != RealDigits[f[m - dm], 10, digits + 2][[1]], m = m + dm; Print[m, " ", RealDigits[f[m]]]];
A0 = f[m]; RealDigits[A0, 10, digits][[1]] (* Jean-François Alcover, Apr 29 2016 *)

default(realprecision, 120); default(parisize, 10000000);
prodeulerrat((1-1/p)*(1 + sum(i = 1, 512, 1/(numbpart(i)*p^i)))) \\ Amiram Eldar, Mar 08 2024

Equals Product_{p prime} (1-Sum_{k >= 2} (1/P(k-1)-1/P(k))/p^k), where P(k) is the unrestricted partition function. - Jean-François Alcover, Apr 29 2016, [typo corrected by Vaclav Kotesovec, Mar 05 2024]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - Amiram Eldar, Oct 16 2020

Data corrected by Jean-François Alcover, Apr 29 2016
a(10) from Vaclav Kotesovec, Mar 07 2024
More terms from Amiram Eldar, Mar 08 2024

cp's OEIS Frontend

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

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A063966 Number of Abelian groups of order <= n.

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A085846 Decimal expansion of root of x = (1+1/x)^x.

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A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

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A084892 Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).

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A084893 Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).

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A104404 Number of groups of order n all of whose subgroups are normal.