OFFSET
0,3
COMMENTS
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/log(x). However, a similar heuristic for the primes fails, as pi(x) ~ x/log(x) and not e^(-gamma)*x/log(x). Here gamma denotes the Euler-Mascheroni constant. - David L. Harden, Aug 24 2002
REFERENCES
David L. Harden, posting to sci.math newsgroup, Jun 06 1999.
LINKS
David Harden, Comments on this sequence
FORMULA
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 L. Harden
EXAMPLE
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.
MATHEMATICA
m = 330;
w[p_] := Product[(x^(p(p^n-1)/(p-1))-1)/(x^((p^n-1)/(p-1))-1), {n, 1, 8}];
T = Select[Table[Exponent[#, x]& /@ List @@ (w[p] + O[x]^m // Normal), {p, Prime[Range[PrimePi[m]]]}], #[[1]] == 0&];
okQ[n_] := AllTrue[T, MemberQ[#, n]&];
Select[Range[0, m], okQ] (* Jean-François Alcover, Nov 08 2019, after David L. Harden *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
EXTENSIONS
More terms from David W. Wilson. Confirmed by David L. Harden, Apr 18, 2002.
STATUS
approved