Throughout this entry, ω, τ, and Ω denote the number of distinct prime factors function, the divisor functionDlmfDlmfMathworldPlanetmath, and the number of (nondistinct) prime factorsMathworldPlanetmath functionMathworldPlanetmath (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction), respectively.


For any positive integer n, 2ω(n)τ(n)2Ω(n).


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:


Hence, 2ω(pk)τ(pk)2Ω(pk). It follows that 2ω(n)τ(n)2Ω(n). ∎

This theorem has an obvious corollary.


For any squarefreeMathworldPlanetmath 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