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A158210
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Number omega(n) of distinct primes dividing n multiplied by -1 when n is squarefree (thus Omega(n) = omega(n)).
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1
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0, -1, -1, 1, -1, -2, -1, 1, 1, -2, -1, 2, -1, -2, -2, 1, -1, 2, -1, 2, -2, -2, -1, 2, 1, -2, 1, 2, -1, -3, -1, 1, -2, -2, -2, 2, -1, -2, -2, 2, -1, -3, -1, 2, 2, -2, -1, 2, 1, 2, -2, 2, -1, 2, -2, 2, -2, -2, -1, 3, -1, -2, 2, 1, -2, -3, -1, 2, -2, -3, -1, 2, -1, -2, 2, 2, -2, -3, -1, 2, 1
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OFFSET
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1,6
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COMMENTS
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This sequence reveals, among the positive integers, which are the unit, the primes, the prime powers, the squarefree (quadratfrei) composites and finally the nonsquarefree composites as per the following:
a(n) < -1: squarefree composites
a(n) = -1: primes (squarefree prime components)
a(n) = 0: unit (squarefree since 1 has no squares of primes as factors)
a(n) = +1: prime powers (nonsquarefree prime components)
a(n) > +1: nonsquarefree composites
The non-squarefree numbers are misleadingly referred to as squareful numbers (squaresome (?) would have been more precise, but this term is not in use).
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..10000
Weisstein, Eric W., Squarefree.
Weisstein, Eric W., Squareful.
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FORMULA
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a(n) = omega(n) * (-1)^mu(n), where mu is the Moebius function.
a(n) = A001221(n) * (-1)^A008683(n).
While omega(n) is additive [i.e. omega(mn) = omega(m) + omega(n), gcd(m,n) = 1], this sequence, while not additive, has the following rule:
a(mn) = [|a(m)| + |a(n)|] * max(sign[a(n)], sign[a(m)]), gcd(m,n) = 1, m > 1, n > 1.
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CROSSREFS
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Cf. A001221 Number of distinct primes dividing n (also called omega(n)).
Cf. A001222 Number of prime divisors of n (counted with multiplicity) (also called Omega(n)).
Cf. A008683 Moebius (or Mobius) function mu(n).
Cf. A005117 Squarefree numbers.
Cf. A013929 Not squarefree numbers.
Cf. A000040 The prime numbers.
Cf. A025475 Powers of a prime but not prime.
Sequence in context: A103765 A125029 A062893 * A087802 A079553 A001221
Adjacent sequences: A158207 A158208 A158209 * A158211 A158212 A158213
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KEYWORD
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sign
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AUTHOR
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Daniel Forgues, Mar 14 2009
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STATUS
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approved
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