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.

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, 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, 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

Offset: 1

Author

Russ Cox, Dec 01 2004

Keywords

• mult
• easy
• nonn

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).

Examples

			The only finite Abelian group of order 6 is C6=C2xC3. The unitary divisors are C1, C2, C3 and C6. So a(6) = 4.
		

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.

Crossrefs

Cf. A000041, A101114.

Cf. A034444, A000688.

Programs

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 *)

PARI

a(n)=my(f=factor(n)[, 2]); prod(i=1, #f, numbpart(f[i]))*2^#f; \\ Michel Marcus, Dec 27 2015

Formula

a(n) = 2^(number of distinct prime factors of n) * product of prime powers in factorization of n.

a(n) = A034444(n) * A000688(n).

Multiplicative with a(p^e) = 2 * A000041(e). - Amiram Eldar, Sep 23 2023