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Search: id:A158522
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| A158522 |
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Dirichlet inverse of number of unitary divisors of n |
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+0
1
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| 1, -2, -2, 2, -2, 4, -2, -2, 2, 4, -2, -4, -2, 4, 4, 2, -2, -4, -2, -4, 4, 4, -2, 4, 2, 4, -2, -4, -2, -8, -2, -2, 4, 4, 4, 4, -2, 4, 4, 4, -2, -8, -2, -4, -4, 4, -2, -4, 2, -4, 4, -4, -2, 4, 4, 4, 4, 4, -2, 8, -2, 4, -4, 2, 4, -8, -2, -4, 4, -8, -2, -4, -2, 4, -4, -4, 4, -8, -2, -4, 2, 4, -2
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Abs{a(n)} = A034444(n). Examples of Dirichlet convolutions with function a(n), i.e. b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A034444(n) = A063524(n), a(n) * A000005(n) = A010052(n), a(n) * A000027(n) = A074722(n), a(n) * A000012(n) = A008836(n).
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FORMULA
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a(n) = (-1)^A001222(n)*A0034444(n) = (-1)^A001222(n)*2^A001221(n), for n >= 2. If n = Product(p_i^e_i), a(n) = Product[(-1)^e_i*2). Multiplicative with a(p_i^e_i) = (-1)^e_i*2. a(1) = 1, a(p) = -2, a(pq) = 4, a(pq...z) = (-1)^k*2^k, a(p^k) = (-1)^k*2 for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).
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EXAMPLE
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a(60) = a(2^2*3*5) = [(-1)^2*2]*[(-1)^1*2]*[(-1)^1*2] = 2*(-2)*(-2) = 8.
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CROSSREFS
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Cf. A034444, A063524, A000005, A010052, A000027, A074722, A001222, A001221, A000040, A006881, A120944, A000961.
Sequence in context: A053238 A058263 A048669 this_sequence A034444 A073180 A127973
Adjacent sequences: A158519 A158520 A158521 this_sequence A158523 A158524 A158525
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KEYWORD
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sign
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AUTHOR
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Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Mar 20 2009
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