Problems & Puzzles: Conjectures

Conjecture 27. A Charles Ashbacher Conjecture

Recently Mr. Charles Ashbacher sent a conjecture related to the well known Smarandache function, S(n). This is his email:

Definition:

For any integer n>=1, the value of the Smarandache function S(n)=m is the smallest integer m >=0 such that n divides m factorial.

It is easy to prove that:

a)S(p) = p, p = prime.
b)S(kp) = p for k < p & p = prime

From this, it follows that for most integers n, S(n) = p where p is a prime. In doing some computations, the percentage of such integers n in the following ranges were as follows:

n range	% n such that S(n) is prime
1 - 10000	0.9406
1 - 20000	0.9526
1 - 30000	0.9587
1 - 40000	0.9623
1 - 50000	0.9649
1 - 60000	0.9669
1 - 70000	0.9685
1 - 100000	0.9719
1 - 200000	0.9775
1 - 500000	0.9832
1 - 750000	0.9852

 

Conjecture: This ratio have a limit of 1.0 as the upper limit of the range goes to infinity?

Questions:

1. Can you add the results for 3 more rows:
1-1000000   ?
1-2000000   ?
1-3000000   ?

2. Can you argument on favor or against this conjecture?

_________
References for the S(n) function:

1. http://mathworld.wolfram.com/SmarandacheFunction.html

Solution