2ω(n)≤τ(n)≤2Ω(n)
Throughout this entry, ω, τ, and Ω denote the number of distinct prime factors function, the divisor function, and the number of (nondistinct) prime factors
function
(http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction), respectively.
Theorem.
For any positive integer n, 2ω(n)≤τ(n)≤2Ω(n).
Proof.
Note that 2ω(n), τ(n), and 2Ω(n) are multiplicative. Also note that, for any positive integer n, the numbers 2ω(n), τ(n), and 2Ω(n) are positive integers. Therefore, it will suffice to prove the inequality for prime powers.
Let p be a prime and k be a positive integer. Thus:
2ω(pk)=2τ(pk)=k+12Ω(pk)=2k
Hence, 2ω(pk)≤τ(pk)≤2Ω(pk). It follows that 2ω(n)≤τ(n)≤2Ω(n). ∎
This theorem has an obvious corollary.
Corollary.
For any squarefree positive integer n, 2ω(n)=τ(n)=2Ω(n).
Title | 2ω(n)≤τ(n)≤2Ω(n) |
---|---|
Canonical name | 2omeganletaunle2Omegan |
Date of creation | 2013-03-22 16:07:28 |
Last modified on | 2013-03-22 16:07:28 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 14 |
Author | Wkbj79 (1863) |
Entry type | Theorem |
Classification | msc 11A25 |
Related topic | NumberOfDistinctPrimeFactorsFunction |
Related topic | TauFunction |
Related topic | NumberOfNondistinctPrimeFactorsFunction |
Related topic | DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2 |