Comments on: What about the composites?

By: bh8153

bh8153 — Sun, 14 Feb 2010 11:37:59 +0000

You are wondering about something related to the omega function, as described in the web page below.

I can see no evidence on this page of investigations of the type you suggest, as to the density distribution and average separation of those numbers which all have the same value of omega, ignoring the numbers with a different value.
References :
http://mathworld.wolfram.com/DistinctPrimeFactors.html

By: knashha

knashha — Sun, 14 Feb 2010 11:12:59 +0000

A great question. The composites per se are virtually ignored.
The composites have a genesis which is pretty obvious:
multiply all the numbers in (2,3,4,5,…..) by all the numbers in (2,3,4,5,…..) and each composite will appear but no primes.
Because of this, the value of the nth composite, A(n)
grows slightly faster than n and the space created is filled
with primes. From this point of view it would seem easier to find the value of the nth composite than to find the value of the nth prime. It is difficult to order the results of the product

(2,3,4,5,…..)(2,3,4,5,…..)

and predict the size of the nth term. If it were possible then
we could find the number of primes not exceeding n to be

A(n) – n – 1 .

This line would be an attempt to understand the growth of the primes without refering to smaller primes but anchoring the
investigation in the global genesis of the composite numbers.
References :

By: thefreevariable

thefreevariable — Sun, 14 Feb 2010 10:26:59 +0000

Prime numbers have lots of interesting properties in algebra and number theory. If I want to study the properties of a composite number, the first thing I want to know is its prime factorization. That’s why composite numbers receive so little attention; they are treated as the product of their prime factors, and that takes us full-circle back to studying primes.

As for the number of divisors for a given composite n, I suppose you could write n in terms of its prime factors and then make consider all possible products of subsets of those factors. That would generate all possible factors.

You might also want to take a look at Euler’s phi-function (sometimes called totient function). I’ve provided a link below. The function, applied to a composite n, gives you the number of integers less than n that share no divisors with n (except 1). Conversely, the cototient of n is the number of integers less than n that share at least one divisor with n.

EDIT: I just read your additional details. That’s a very interesting question in my opinion. I don’t know the answer, but if you’re really into number theory, perhaps you could solve it and enlighten us!
References :
http://en.wikipedia.org/wiki/Euler_phi_function

By: brilliantwaz

brilliantwaz — Sun, 14 Feb 2010 09:48:59 +0000

let us take 8.
8=2^3
number of factors of 8 are 1,2,4,8 which are 4.
we can observe that it is 3+1=4.
take 12
12=2^2*3
factors of 12 are 1,2,3,4,6,12 which are 6.
we can observe that it has (2+1)(1+1)=6 divisors.
there is a theorem in number theory that a composite number greater then 2,can be written as product of prime numbers . suppose a number can be written as a^x*b^y*c^z, where a,b,c are prime factors of the number and x,y,z are their powers, then the number of divisors or factors of the number are (x+1)(y+1)(z+1).the above examples may clear this point and may help u in ur research.
References :

By: arty.fishal

arty.fishal — Sun, 14 Feb 2010 09:42:59 +0000

The nth composite is very easy to determine. It is at least 1 + the nth-1 composite, and at most two. The composites are not the subject of much investigation, for their pattern is obvious and unlike the primes, their distribution does not give insight into the overall pattern of the integers. Furthermore it is unknown if there is a formula for predicting the nth prime, it is still under investigation.
References :