OFFSET
1,2
COMMENTS
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).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Peter Georg Schmidt, Zur Anzahl unitärer Faktoren abelscher Gruppen [On the number of unitary factors in Abelian groups], Acta Arith., 64 (1993), 237-248.
J. Wu, On the average number of unitary factors of finite Abelian groups, Acta Arith. 84 (1998), 17-29.
FORMULA
a(n) = 2^(number of distinct prime factors of n) * product of prime powers in factorization of n.
Multiplicative with a(p^e) = 2 * A000041(e). - Amiram Eldar, Sep 23 2023
EXAMPLE
The only finite Abelian group of order 6 is C6=C2xC3. The unitary divisors are C1, C2, C3 and C6. So a(6) = 4.
MATHEMATICA
Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[n]]]]
Table[2^(PrimeNu[n])*(Times @@ PartitionsP /@ Last /@ FactorInteger@n), {n, 1, 100}] (* G. C. Greubel, Dec 27 2015 *)
PROG
(PARI) a(n)=my(f=factor(n)[, 2]); prod(i=1, #f, numbpart(f[i]))*2^#f; \\ Michel Marcus, Dec 27 2015
CROSSREFS
KEYWORD
mult,easy,nonn
AUTHOR
Russ Cox, Dec 01 2004
STATUS
approved