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%I #23 Oct 01 2023 02:41:08
%S 1,2,1,1,4,1,1,2,1,2,1,6,3,2,1,1,4,1,1,1,2,2,1,1,10,1,5,1,1,4,4,1,2,1,
%T 1,6,1,1,3,2,5,4,1,1,2,1,1,2,1,14,1,2,2,1,9,1,1,1,2,1,1,6,4,1,2,1,1,1,
%U 1,4,3,2,1,2,10,3,1,5,1,1,4,1,8,1,6,3,1,2,1,1,4,1,6,1,1,2,2,3,21,1,1,2,1,2,4,1,1,1,2
%N Number of non-cyclic abelian groups of finite order. The order is given by A013929.
%C Every abelian group of finite order is the direct product of cyclic groups (there may be only one factor). See, e.g., the A. Speiser reference, Satz 43, p. 49, in combination with Satz 42, p. 47, and also Satz 4, p. 17, with the remark on the direct product on page 28.
%C See the list of abelian groups of small order in the Wikipedia link.
%D Andreas Speiser, Die Theorie der Gruppen von endlicher Ordnung, Vierte Auflage, Birkhäuser, 1956.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_small_groups">List of small groups</a>.
%F a(n) = A000688(A013929(n)) - 1, n>=1.
%F See the formula for A000688 using the product of the number of partitions of the exponents in the prime number factorization.
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2) * c - 1)/(zeta(2) - 1) - 1 = 3.3025914257..., where c = A021002. - _Amiram Eldar_, Oct 01 2023
%e n=1: there is one abelian group of order 4=A013929(1), which is not the cyclic group Z_4 (in additive notation), namely the Klein 4-group: Z_2 x Z_2, (also denoted by (Z_2)^2.
%e n=2: there are 2 non-cyclic abelian groups of order 8=A013929(2), namely Z_2 x Z_4 and (Z_2)^3.
%e n=3: order 9=A013929(3), (Z_3)^2.
%e n=4: order 12, Z_3 x (Z_2)^2 (note that Z_6 = Z_3 x Z_2 and Z_12 = Z_4 x Z_3, where = means 'is isomorphic to').
%e n=5: order 16. The four non-cyclic groups are (Z_2)^4, Z_4 x (Z_2)^2, Z_8 x Z_2 and (Z_4)^2.
%t FiniteAbelianGroupCount /@ Select[Range[300], ! SquareFreeQ[#] &] - 1 (* _Amiram Eldar_, Oct 01 2023 *)
%Y Cf. A000688, A013929, A013661, A021002.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jul 28 2011
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