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Search: id:A000688
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| A000688 |
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Number of factorizations of n into prime powers greater than 1; number of Abelian groups of order n.
(Formerly M0064 N0020)
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+0
33
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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; listen)
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OFFSET
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1,4
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COMMENT
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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
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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.
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LINKS
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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, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to groups
Index entries for "core" sequences
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FORMULA
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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.
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MAPLE
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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
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f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v Sep 22 2006 *)
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CROSSREFS
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Cf. A000001, A000041, A000961, A001055, A034382, A046054, A046055, A046056, A050360.
Cf. A055653.
Adjacent sequences: A000685 A000686 A000687 this_sequence A000689 A000690 A000691
Sequence in context: A005361 A008479 A107345 this_sequence A038538 A088529 A136565
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KEYWORD
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nonn,core,easy,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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