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Search: id:A158523
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| A158523 |
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Moebius transform of negate primes in factorization of n. |
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1
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| 1, -3, -4, 6, -6, 12, -8, -12, 12, 18, -12, -24, -14, 24, 24, 24, -18, -36, -20, -36, 32, 36, -24, 48, 30, 42, -36, -48, -30, -72, -32, -48, 48, 54, 48, 72, -38, 60, 56, 72, -42, -96, -44, -72, -72, 72, -48, -96, 56, -90, 72, -84, -54, 108, 72, 96, 80, 90, -60, 144
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OFFSET
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1,2
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COMMENT
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a(n) = mu(n) * A061019(n). = A008683(n) * A061019(n). a(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution for n >= 1. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d). Inverse Moebius transform of a(n) is A061019(n). Abs[a(n)] is Dedekind psi function A001615(n). a(n) = (-1)^A001222(n)*A001615(n). Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1).
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FORMULA
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Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1). a(1)=1, a(p)=(-1)*(p+1), a(pq)=(p+1)*(q+1), a(pq...z)=(-1)^k*(p+1)*(q+1)*...*(z+1), a(p^k)=(-1)^k*(p+1)*p^(e-1), 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(72)=a(2^3*3^2)=[(-1)^3*(2+1)*2^(3-1)]*[(-1)^2*(3+1)*3^(2-1)]=(-12)*12=-144.
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CROSSREFS
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Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961.
Sequence in context: A023830 A063649 A053158 this_sequence A001615 A133689 A135510
Adjacent sequences: A158520 A158521 A158522 this_sequence A158524 A158525 A158526
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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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