%I #17 Nov 08 2019 05:58:05
%S 0,1,4,8,10,18,22,26,32,34,46,49,50,57,66,70,74,81,82,86,94,102,130,
%T 134,138,142,152,162,165,166,174,176,183,184,201,205,206,222,231,232,
%U 236,237,244,246,256,270,273,274,286,290,296,304,312,318,326
%N Every prime occurs to this power in some factorial.
%C 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
%D David L. Harden, posting to sci.math newsgroup, Jun 06 1999.
%H David Harden, <a href="/A048247/a048247.txt">Comments on this sequence</a>
%F 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_
%e Given any prime p, there exists a positive integer n such that p|n! but p^2 does not divide n!.
%e Given any prime p, there exists a positive integer n such that p^4|n! but p^5 does not divide n!.
%e 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!).
%e 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.
%t m = 330;
%t w[p_] := Product[(x^(p(p^n-1)/(p-1))-1)/(x^((p^n-1)/(p-1))-1), {n, 1, 8}];
%t T = Select[Table[Exponent[#, x]& /@ List @@ (w[p] + O[x]^m // Normal), {p, Prime[Range[PrimePi[m]]]}], #[[1]] == 0&];
%t okQ[n_] := AllTrue[T, MemberQ[#, n]&];
%t Select[Range[0, m], okQ] (* _Jean-François Alcover_, Nov 08 2019, after _David L. Harden_ *)
%K nonn,easy,nice
%O 0,3
%A Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
%E More terms from _David W. Wilson_. Confirmed by _David L. Harden_, Apr 18, 2002.