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A048247
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Every prime occurs to this power in some factorial.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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/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
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REFERENCES
| David Harden (w_harden(AT)bellsouth.net), posting to sci.math newsgroup, Jun 06 1999.
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LINKS
| David Harden, Comments on this sequence
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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 Harden (sylow2subgroup(AT)hotmail.com).
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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.
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CROSSREFS
| Sequence in context: A046559 A032618 A090696 * A130442 A031073 A037004
Adjacent sequences: A048244 A048245 A048246 * A048248 A048249 A048250
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net). Confirmed by David Harden, Apr 18, 2002.
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