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

Jud McCranie extended the Ashbacher's table far beyond the ranges asked. This is only a small part of his  calculations:

1000000 0.9864320000
2000000 0.9890445000
3000000 0.9903273333
4000000 0.9911412500
5000000 0.9917168000
6000000 0.9921600000
7000000 0.9925180000
8000000 0.9928091250
9000000 0.9930598889
10000000 0.9932747000
...
347000000 0.9975904582
348000000 0.9975923218
349000000 0.9975943381
350000000 0.9975962686

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But the same Jud discovered that another version of this conjecture was posted by Erdös and solved by I. Kastanas, according to a Web page posted by Steven Finch. There you can read:

...let P(n) denote the largest prime factor of n...

Erdös [4,5] pointed out that for almost all n, meaning

as N approaches infinity. Kastanas [5] proved this to be true, hence the following argument is valid.

...

1. P. Erdös, personal communication to T. Yau (1995), in Smarandache Notions Journal 8 (1997) 220-224.
2. I. Kastanas, The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.

Does anybody may send to our pages the Kastanas proof?

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