A101114 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 <= n.

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 41, 43, 47, 51, 61, 63, 71, 73, 81, 85, 89, 91, 103, 107, 111, 117, 125, 127, 135, 137, 151, 155, 159, 163, 179, 181, 185, 189, 201, 203, 211, 213, 221, 229, 233, 235, 255, 259, 267, 271, 279, 281, 293, 297, 309, 313, 317

Offset: 1

Author

Russ Cox, Dec 01 2004

Keywords

• 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. Sequence gives T(n) = sum_{G in A, |G| <= n} t(G).

Examples

			A101113 begins 1, 2, 2, 4, 2. So a(5) = 11.
		

References

• Schmidt, Peter Georg, Zur Anzahl unitaerer Faktoren abelscher Gruppen. [On the number of unitary factors in Abelian groups] Acta Arith., 64 (1993), 237-248.
• Wu, J., On the average number of unitary factors of finite Abelian groups, Acta Arith. 84 (1998), 17-29.

Crossrefs

Cf. A101113.

Programs

Mathematica

Sum[Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[i]]]], {i, n}]

Formula

a(n) = partial sums of A101113