A048247 Every prime occurs to this power in some factorial.

0, 1, 4, 8, 10, 18, 22, 26, 32, 34, 46, 49, 50, 57, 66, 70, 74, 81, 82, 86, 94, 102, 130, 134, 138, 142, 152, 162, 165, 166, 174, 176, 183, 184, 201, 205, 206, 222, 231, 232, 236, 237, 244, 246, 256, 270, 273, 274, 286, 290, 296, 304, 312, 318, 326

Offset: 0

Author

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

Keywords

• nonn
• easy
• nice

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

Examples

			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.
		

References

• David L. Harden, posting to sci.math newsgroup, Jun 06 1999.

Links

• David Harden, Comments on this sequence

Programs

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 *)

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

Extensions

More terms from David W. Wilson. Confirmed by David L. Harden, Apr 18 2002.