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Search: id:A101114
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| A101114 |
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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. |
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2
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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).
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REFERENCES
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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.
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FORMULA
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a(n) = partial sums of A101113
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EXAMPLE
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A101113 begins 1, 2, 2, 4, 2. So a(5) = 11.
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MATHEMATICA
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Sum[Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[i]]]], {i, n}]
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CROSSREFS
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Cf. A101113.
Adjacent sequences: A101111 A101112 A101113 this_sequence A101115 A101116 A101117
Sequence in context: A024896 A040976 A078651 this_sequence A120696 A071156 A076610
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KEYWORD
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easy,nonn
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AUTHOR
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Russ Cox (rsc(AT)swtch.com), Dec 01 2004
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