0,3

There are no primes in the sequence, as the prime p fails the base p test. The set of positive integers failing the base p test for membership has density 1/p. Also, when n is a nonmember of the set, any base p whose test n fails has p<=n. Therefore one conjectural estimate for the number of members of the set <=x would be x*product{primes p<=x}(1-1/p) ~ e^(-gamma)*x/ln(x). However, a similar heuristic for the primes fails, as pi(x) ~ x/ln(x) and not e^(-gamma)*x/ln(x). Here gamma denotes the Euler-Mascheroni constant. - David Harden (sylow2subgroup(AT)hotmail.com), Aug 24 2002

David Harden (w_harden(AT)bellsouth.net), posting to sci.math newsgroup, Jun 06 1999.

David Harden, Comments on this sequence

Numbers passing the test for membership for the base p are generated by W_p(x) = product_{n=1..inf} (x^(p*(p^n-1)/(p-1))-1)/(x^((p^n-1)/(p-1))-1) - David Harden (sylow2subgroup(AT)hotmail.com).

Given any prime p, there exists a positive integer n such that p|n! but p^2 does not divide n!.

Given any prime p, there exists a positive integer n such that p^4|n! but p^5 does not divide n!.

But it is not true that given any prime p, there exists a positive integer n such that p^6|n! but p^7 does not divide n! (for if 64|n! then 128|n!).

For every prime p there is an n such that p^4|n! but p^5 doesn't divide n!: for p=2, we may take n=6; for p=3, we may take n=9; for p>4, we may take n=4p.

Sequence in context: A046559 A032618 A090696 this_sequence A130442 A031073 A037004

Adjacent sequences: A048244 A048245 A048246 this_sequence A048248 A048249 A048250

nonn,easy,nice

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)

More terms from David W. Wilson (davidwwilson(AT)comcast.net). Confirmed by David Harden, Apr 18, 2002.