1,6

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).

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

Weisstein, Eric W., Squarefree.

Weisstein, Eric W., Squareful.

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.

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 Square-free numbers.

Cf. A013929 Not square-free numbers.

Cf. A000040 The prime numbers.

Cf. A025475 Powers of a prime but not prime.

Sequence in context: A103765 A125029 A062893 this_sequence A087802 A079553 A001221

Adjacent sequences: A158207 A158208 A158209 this_sequence A158211 A158212 A158213

sign

Daniel Forgues (squid(AT)zensearch.com), Mar 14 2009