Number Theory and Probability

[Ruffgos]

What is the density of numbers with only consequitive prime factors starting with 2 (for example, 6 = 2×3 (first times second prime), 60 = 2²×3×5, etc., but not 7×11 or 2×5×7, etc.) among all natural numbers?

Alternatively, if the density is 0, what would be the estimate of the percentage of the ammount of such numbers among naturals up to some N?

 

[Deleted member 4993]

What is the density of numbers with only consequitive prime factors starting with 2 (for example, 6 = 2×3 (first times second prime), 60 = 2²×3×5, etc., but not 7×11 or 2×5×7, etc.) among all natural numbers?

Alternatively, if the density is 0, what would be the estimate of the percentage of the ammount of such numbers among naturals up to some N?

What is the definition of "density of numbers" ?

Please share your work/thoughts about this problem.

 

[blamocur]

What is the density of numbers with only consequitive prime factors starting with 2 (for example, 6 = 2×3 (first times second prime), 60 = 2²×3×5, etc., but not 7×11 or 2×5×7, etc.) among all natural numbers?

Alternatively, if the density is 0, what would be the estimate of the percentage of the ammount of such numbers among naturals up to some N?

Are you talking about all even numbers by chance?

 

[BigBeachBanana]

What is the density of numbers with only consequitive prime factors starting with 2 (for example, 6 = 2×3 (first times second prime), 60 = 2²×3×5, etc., but not 7×11 or 2×5×7, etc.) among all natural numbers?

Alternatively, if the density is 0, what would be the estimate of the percentage of the ammount of such numbers among naturals up to some N?

A number can always be expressed as a product of its prime factors. So if the prime factorization has a factor of 2, then it must be an even number. Second, you said it has to start with 2, but multiplication is commutative. In your example, 2x3=3x2.

 

[Dr.Peterson]

What is the density of numbers with only consequitive prime factors starting with 2 (for example, 6 = 2×3 (first times second prime), 60 = 2²×3×5, etc., but not 7×11 or 2×5×7, etc.) among all natural numbers?

Alternatively, if the density is 0, what would be the estimate of the percentage of the ammount of such numbers among naturals up to some N?

Are you saying the numbers must each be the product of all consecutive primes up to some value, e.g. {2, 3, 5, 7}, each to a non-zero power?

And how do you want to describe "density"? The limit of the fractions of such numbers up to N, as N goes to infinity, perhaps?

Do you have reason to think this is possible, or is it just a matter of curiosity? It is possible to ask a question like this that can take many great minds to solve; or it may turn out to be easy.

 

[blamocur]

Are you talking about all even numbers by chance?

Ooops, did not pay attention to the consecutive part.
(Thank you @Cubist).

 

[Cubist]

I wrote some code that counts the consecutive-prime-factor-numbers up to a defined maximum

maximum	#consecutive-prime-factor-numbers	density (#consecutive-prime-factor-numbers / maximum)
10	4	0.4
100	18	0.18
1,000	56	0.056
10,000	147	0.0147
100,000	350	0.0035
1,000,000	784	0.000784
10,000,000	1660	0.000166
100,000,000	3377	0.00003377
1,000,000,000	6656	0.000006656

Spoiler: show code

from sympy import nextprime

def upto(maxn, n, factor):
    count = 1
    # print(n) # uncomment to display all the consecutive-prime-factor-numbers
    np = nextprime(factor)
    while n*factor <= maxn:
        n *= factor
        count += upto(maxn, n, np)
    return count

#######################

maxn=100000
count=upto(maxn, 1, 2)
count -= 1 # discard the result for "1"
print("#consecutive-prime-factor-numbers up to",maxn,"=",count)