Weird Sequence of Positive Integers (FWD)

[Antreas P. Hatzipolakis]

>From: w_harden at bellsouth.net (David Harden)
>Newsgroups: sci.math
>Subject: Weird Sequence of Positive Integers
>Date: Sun, 06 Jun 1999 19:16:08 GMT
>
>
>The following statement is trivial:
>
>Given any prime p, there exists a positive integer n such that p|n!
>but p^2 does not divide n!.
>
>The following statement is also true, and takes a little checking:
>
>Given any prime p, there exists a positive integer n such that p^4|n!
>but p^5 does not divide n!.
>
>As does this one:
>
>Given any prime p, there exists a positive integer n such that p^8|n!
>but p^9 does not divide n!.
>
>And this one:
>
>Given any prime p, there exists a positive integer n such that p^10|n!
>but p^11 does not divide n!.
>
>But not this one:
>
>Given any prime p, there exists a positive integer n such that p^6|n!
>but p^7 does not divide n!.
>
>For a counterexample, note that if 64 divides n!, 128 also does. p=2
>is the only counterexample to this particular statement.
>
>In general, for what positive integers k does there always exist a
>positive integer n such that for any prime p p^k divides n! but
>p^(k+1) does not? The first few values of k like this are 1, 4, 8, 10,
>18, 22, 26, 32, 34, 46, 49, 50, 57, 66, 70, 74, 81, 82, 94, 102, 130,
>134, 138, 142, 152, 162, 165, 166, 174, 176, 183, 184, 201, 205, 206,
>222, 231, 232, 236, 237, 244, 246, and 256.
>
>It is easy to show that k is in this set if and only if, for every
>prime p it can be expressed as a sum of the form c_r*((p^r-1)/(p-1))
>for nonnegative integers r and 0<=c_r<p. In general, the density of
>positive integers which have a given prime p as a counterexample that
>shows that they are not in the set in the set of all positive integers
>is 1/p, and the positive integers passing this test for the base p
>have density 1-1/p. Therefore I expect heuristically that this set is
>infinite (Can someone out there prove this?) and that the nth term of
>this sequence is asymptotically equivalent to the nth prime (even
>harder!).
>
>Note that the set of positive integers passing the test for membership
>in this sequence for the base p can be generated by the function
>
>W_p(x)=the product of ((x^p*((p^n-1)/(p-1))-1)/(x-1)) from n=1 to
>infinity
>
>and that therefore those failing the test can be generated by the
>function
>
>1/(1-x)-W_p(x).
>
>The sum of 1/(1-x)-W_p(x) over all primes p converges for all x with
>|x|<1 because the coefficient of x^n is bounded by pi(n) since n
>passes the test for the base p if p>n.
>
>I don't know if this helps, but it would be interesting to see these
>questions answered.
>
>---- David Harden