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A158210 Number omega(n) of distinct primes dividing n multiplied by -1 when n is squarefree (thus Omega(n) = omega(n)). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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 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 nonsquarefree numbers are misleadingly referred to as squareful numbers (squaresome (?) would have been more precise, but this term is not in use).

LINKS

Daniel Forgues, Table of n, a(n) for n=1..10000

Weisstein, Eric W., Squarefree.

Weisstein, Eric W., Squareful.

FORMULA

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.

MATHEMATICA

Table[(-1)^MoebiusMu[n]*PrimeNu[n], {n, 81}] (* L. Edson Jeffery, Dec 08 2014 *)

CROSSREFS

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 Nonsquarefree numbers.

Cf. A000040 The prime numbers.

Cf. A025475 Powers of a prime but not prime.

Sequence in context: A237353 A293460 A231813 * A087802 A079553 A001221

Adjacent sequences:  A158207 A158208 A158209 * A158211 A158212 A158213

KEYWORD

sign

AUTHOR

Daniel Forgues, Mar 14 2009

STATUS

approved

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Last modified December 14 00:21 EST 2017. Contains 295976 sequences.