For any integer n ≥ 1, let

d(n) = number of positive divisors of n

v(n) = number of distinct prime divisors of n

ω(n) = number of prime divisors of n counted with multiplicity

[for example : If p is prime, then d(p) = 2, v(p) = v(p²) = 1, \(ω\)(p²) = 2]

if n ≥ 1000 and \(\omega\)(n) > 2, then d(n) > log n

there exists n such that d(n) > 3\(\sqrt{n}\)

for every n, 2^v(n) ≤ d(n) ≤ 2^{\(\omega\)(n)}

if \(\omega\)(n) = \(\omega\)(m), then d(n) = d(m)

For any integer n ≥ 1, let d(n) = number of positive divisors of n v(n) = number of distinct prime divisors of n ω(n) = number of prime divisors of n counted with multiplicity [for example : If p is prime, then d(p) = 2, v(p) = v(p2) = 1, \(ω\)(p2) = 2]