%I A000688 M0064 N0020
%S A000688 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,
%T A000688 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,
%U A000688 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
%N A000688 Number of factorizations of n into prime powers greater than 1; number 
               of Abelian groups of order n.
%C A000688 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).
%C A000688 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
%C A000688 Range is A033637.
%D A000688 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.
%D A000688 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
%D A000688 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, 
               p. 468.
%D A000688 H.-E. Richert, Ueber die Anzahl Abelscher Gruppen gegebener Ordnung I, 
               Math. Zeitschr. 56 (1952) 21-32.
%D A000688 J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 
               7.
%D A000688 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000688 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000688 T. D. Noe, <a href="b000688.txt">Table of n, a(n) for n = 1..10000</a>
%H A000688 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/abel/abel.html">
               Abelian Group Enumeration Constants</a>
%H A000688 B. Horvat, G. Jaklic and T. Pisanski, <a href="http://arXiv.org/abs/math.CO/
               0503183">On the number of Hamiltonian groups</a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AbelianGroup.html">Link to a section of The World of Mathematics.</
               a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FiniteGroup.html">Link to a section of The World of Mathematics.</
               a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KroneckerDecompositionTheorem.html">Link to a section of The World 
               of Mathematics.</a>
%H A000688 <a href="Sindx_Gre.html#groups">Index entries for sequences related to 
               groups</a>
%H A000688 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000688 a(p^k) = number of partitions of k; a(mn)=a(m)a(n) if (m, n)=1.
%F A000688 Multiplicative with a(p^e) = A000041(e). - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%p A000688 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
%t A000688 f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] 
               (* Robert G. Wilson v Sep 22 2006 *)
%Y A000688 Cf. A000001, A000041, A000961, A001055, A034382, A046054, A046055, A046056, 
               A050360.
%Y A000688 Cf. A055653.
%Y A000688 Sequence in context: A005361 A008479 A107345 this_sequence A038538 A088529 
               A136565
%Y A000688 Adjacent sequences: A000685 A000686 A000687 this_sequence A000689 A000690 
               A000691
%K A000688 nonn,core,easy,nice,mult
%O A000688 1,4
%A A000688 N. J. A. Sloane (njas(AT)research.att.com).