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A129667
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Dirichlet inverse of the Abelian group count (A000688).
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0
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1, -1, -1, -1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 0, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 0, -1, -1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, 0, 1, 0, 1, 1, -1, -1, -1, 1, 1, 0, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 1, 1, 1, -1, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 0, -1, 1, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| The simple formula which gives the value of this multiplicative function for the power of any prime can be derived from Euler's celebrated "Pentagonal Number Theorem" (applied to the generating function of the partition function A000041 on which A000688 is based).
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LINKS
| G. P. Michon, Partition Function and Pentagonal Numbers.
G. P. Michon, Multiplicative Functions.
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FORMULA
| Multiplicative function for which a(p^e) either vanishes or is equal to (-1)^n, for any prime p, if e is either n(3n-1)/2 or n(3n+1)/2 (these integers are the pentagonal numbers of the first and second kind, A000326 and A005449).
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EXAMPLE
| a(8) and a(27) are zero because the sequence vanishes for the cubes of primes. Not so with fifth powers of primes (since 5 is a pentagonal number) so a(32) is nonzero.
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CROSSREFS
| Cf. A000041, A000326, A000688, A005449, A023900, A101035.
Sequence in context: A119981 A115789 A053864 * A071374 A071025 A077010
Adjacent sequences: A129664 A129665 A129666 * A129668 A129669 A129670
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KEYWORD
| mult,easy,sign
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AUTHOR
| Gerard P. Michon (g.michon(AT)att.net), Apr 28 2007, May 01 2007
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