- In a message dated 05/05/03 10:25:14 GMT Daylight Time, ambroxius@...

writes:

> Which is the best manner to calculate the average of factors of the

It tends to infinity.

> numbers up to a fixed N? And between N and M?

>

> (for example, up to 10: 1,2...,9,10 have 1,1,1,2,1,2,1,3,2,2 factors, the

> average is 1,6)

>

> Is the average tending to infinite or to a specific ratio?

>

The standard number theory function you are interested in is d(n), defined to

be the number of divisors of n, including 1 and n, so that d(n) >= 2.

Then Hardy and Wright "An Introduction to the Theory of Numbers" (1979) (p.

264) have :-

"Theorem 319. The average order of d(n) is log n."

An alternative formulation of this result is:-

lim {n -> oo} [d(1) + d(2) + ... + d(n)] / log(n) = 1.

Mike Oakes

[Non-text portions of this message have been removed] - Sent too quickly - I should have written:

lim {n -> oo} [d(1) + d(2) + ... + d(n)] / (n*log(n)) = 1.

Mike

[Non-text portions of this message have been removed] - Hi:

I was thinking on prime factors rather than in divisors... the number of them is quite lesser than this of the divisors... what order has?

Jose----- Original Message -----

From: mikeoakes2@...

To: primenumbers@yahoogroups.com

Sent: Monday, May 05, 2003 1:39 PM

Subject: Re: [PrimeNumbers] Number of factors in average

In a message dated 05/05/03 10:25:14 GMT Daylight Time, ambroxius@...

writes:

> Which is the best manner to calculate the average of factors of the

> numbers up to a fixed N? And between N and M?

>

> (for example, up to 10: 1,2...,9,10 have 1,1,1,2,1,2,1,3,2,2 factors, the

> average is 1,6)

>

> Is the average tending to infinite or to a specific ratio?

>

It tends to infinity.

The standard number theory function you are interested in is d(n), defined to

be the number of divisors of n, including 1 and n, so that d(n) >= 2.

Then Hardy and Wright "An Introduction to the Theory of Numbers" (1979) (p.

264) have :-

"Theorem 319. The average order of d(n) is log n."

An alternative formulation of this result is:-

lim {n -> oo} [d(1) + d(2) + ... + d(n)] / log(n) = 1.

Mike Oakes

[Non-text portions of this message have been removed]

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[Non-text portions of this message have been removed] - In a message dated 06/05/03 00:40:56 GMT Daylight Time, ambroxius@...

writes:

> I was thinking on prime factors rather than in divisors... the number of

them is quite

> lesser than this of the divisors... what order has?

Sorry, my fault.

That standard number theory function is Omega(n), defined to be the total

number of prime factors of n; in other words, if there is the prime

factorisation

n = p_1^e_1 * ... * p_r^e_r,

then

Omega(n) = e_1 + ... + e_r.

So, in particular Omega(1) = 0. [As an aside: anyone who thinks 1 is a prime

would have a hard job defining Omega(); and 1 is certainly not composite...]

Omega(n) has average order log(log(n)).

Mike

[Non-text portions of this message have been removed]