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A000688
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Number of Abelian groups of order n; number of factorizations of n into prime powers.
(Formerly M0064 N0020)
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40
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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 (FrankTAW(AT)Netscape.net), Oct 20 2006
Range is A033637.
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REFERENCES
| P. Erdos and G. Szekeres, Ueber die Anzahl der Abelschen Gruppen gegebener Ordnung und ueber ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
H.-E. Richert, Ueber die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
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, Birkhaeuser, 1956.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
S. R. Finch, Abelian Group Enumeration Constants
B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups
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
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FORMULA
| a(p^k) = number of partitions of k; a(mn)=a(m)a(n) if (m, n)=1.
Multiplicative with a(p^e) = A000041(e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
a(n) = product(pa(e(j)),j=1..N(n)), n>=2, if
n = product(p(j)^e(j),j=1..N(n)) with p(j) = A000040(j) (primes), N(n) = A001221(n)(also called omega(n)), and pa(n) = A000041(n) (partition numbers). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - Wolfdieter Lang, Jul 23 2011.
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EXAMPLE
| 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)
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MAPLE
| 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: # from James A. Sellers Dec 07 2000
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MATHEMATICA
| f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v Sep 22 2006 *)
Table[FiniteAbelianGroupCount[n], {n, 200}] (* ver.7 *) (* From Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
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PROG
| (PARI) A000688(n) = {local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))} [From Michael B. Porter (michael_b_porter(AT)yahoo.com), Feb 08 2010]
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CROSSREFS
| Cf. A000001, A000041, A000961, A001055, A034382, A046054, A046055, A046056, A050360.
Cf. A055653.
Sequence in context: A005361 A008479 A107345 * A038538 A088529 A136565
Adjacent sequences: A000685 A000686 A000687 * A000689 A000690 A000691
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KEYWORD
| nonn,core,easy,nice,mult,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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