%I #18 Sep 23 2023 03:41:15
%S 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,
%T 4,4,16,2,4,4,12,2,8,2,8,8,4,2,20,4,8,4,8,2,12,4,12,4,4,2,16,2,4,8,22,
%U 4,8,2,8,4,8,2,24,2,4,8,8,4,8,2,20,10,4,2,16,4,4,4,12,2,16,4,8,4,4,4,28,2,8
%N 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 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).
%H G. C. Greubel, <a href="/A101113/b101113.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Georg Schmidt, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6433.pdf">Zur Anzahl unitärer Faktoren abelscher Gruppen</a> [On the number of unitary factors in Abelian groups], Acta Arith., 64 (1993), 237-248.
%H J. Wu, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa84/aa8412.pdf">On the average number of unitary factors of finite Abelian groups</a>, Acta Arith. 84 (1998), 17-29.
%F a(n) = 2^(number of distinct prime factors of n) * product of prime powers in factorization of n.
%F a(n) = A034444(n) * A000688(n).
%F Multiplicative with a(p^e) = 2 * A000041(e). - _Amiram Eldar_, Sep 23 2023
%e 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 Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[n]]]]
%t Table[2^(PrimeNu[n])*(Times @@ PartitionsP /@ Last /@ FactorInteger@n), {n, 1, 100}] (* _G. C. Greubel_, Dec 27 2015 *)
%o (PARI) a(n)=my(f=factor(n)[, 2]); prod(i=1, #f, numbpart(f[i]))*2^#f; \\ _Michel Marcus_, Dec 27 2015
%Y Cf. A000041, A101114.
%Y Cf. A034444, A000688.
%K mult,easy,nonn
%O 1,2
%A _Russ Cox_, Dec 01 2004