%I A101113
%S A101113 1,2,2,4,2,4,2,6,4,4,2,8,2,4,4,10,2,8,2,8,4,4,2,12,4,4,6,8,2,8,2,14,4,
               4,
%T A101113 4,16,2,4,4,12,2,8,2,8,8,4,2,20,4,8,4,8,2,12,4,12,4,4,2,16,2,4,8,22,4,
               8,
%U A101113 2,8,4,8,2,24,2,4,8,8,4,8,2,20,10,4,2,16,4,4,4,12,2,16,4,8,4,4,4,28,2,
               8
%N A101113 Let t(G) = number of unitary factors of the Abelian group G. Then a(n) 
               = sum t(G) over all Abelian groups G of order exactly n.
%C A101113 From Schmidt paper: Let A denote the set of all Abelian groups. Under 
               the operation of direct product, A is a semigroup with identity element 
               E, the group with one element. G_1 and G_2 are relatively prime if 
               the only common direct factor of G_1 and G_2 is E. We say that G_1 
               and G_2 are unitary factors of G if G=G_1 X G_2 and G_1, G_2 are 
               relatively prime. Let t(G) denote the number of unitary factors of 
               G. This sequence is a(n) = sum_{G in A, |G| = n} t(n).
%C A101113 A101113(n) = A034444(n) * A000688(n).
%D A101113 Schmidt, Peter Georg, Zur Anzahl unitaerer Faktoren abelscher Gruppen. 
               [On the number of unitary factors in Abelian groups] Acta Arith., 
               64 (1993), 237-248.
%D A101113 Wu, J., On the average number of unitary factors of finite Abelian groups, 
               Acta Arith. 84 (1998), 17-29.
%F A101113 a(n) = 2^(number of distinct prime factors of n) * product of prime powers 
               in factorization of n.
%e A101113 The only finite Abelian group of order 6 is C6=C2xC3. The unitary divisors 
               are C1, C2, C3 and C6. So a(6) = 4.
%t A101113 Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[n]]]]
%Y A101113 Cf. A101114.
%Y A101113 Cf. A034444, A000688.
%Y A101113 Sequence in context: A147848 A129089 A124315 this_sequence A055155 A085191 
               A165872
%Y A101113 Adjacent sequences: A101110 A101111 A101112 this_sequence A101114 A101115 
               A101116
%K A101113 mult,easy,nonn
%O A101113 1,2
%A A101113 Russ Cox (rsc(AT)swtch.com), Dec 01 2004