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A057109
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Numbers n which are not a factor of P(n)!, where P(n) is the largest prime factor of n.
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11
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4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 45, 48, 49, 50, 54, 64, 72, 75, 80, 81, 90, 96, 98, 100, 108, 121, 125, 128, 135, 144, 147, 150, 160, 162, 169, 175, 180, 189, 192, 196, 200, 216, 224, 225, 240, 242, 243, 245, 250, 256, 270, 288, 289, 294, 300, 320, 324
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OFFSET
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1,1
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COMMENTS
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These are also the numbers for which the Kempner function A002034 is composite. Their density approaches zero as they go to infinity. - Jud McCranie, Dec 08 2001
n is a member if and only if P(n) < A002034(n). The members are the exceptions to the rule that P(n) = A002034(n) for almost all n (Erdos and Kastanas 1994, Ivic 2004). - Jonathan Sondow, Jan 10 2005
Same as numbers n such that |e - m/n| < 1/(P(n)+1)! for some integer m. - Jonathan Sondow, Dec 29 2007
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284-292.
P. Erdos and I. Kastanas, Problem/Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
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LINKS
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Table of n, a(n) for n=1..58.
S. R. Finch, The Average Value of the Smarandache Function
C. Rivera, Conjecture about their density
Eric Weisstein's World of Mathematics, Smarandache function
A. Ivic (2004), On a problem of Erdos involving the largest prime factor of n
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality
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EXAMPLE
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12 is in the sequence since 3 is the largest prime factor of 12, but 12 is not a factor of 3!=6.
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MAPLE
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with(numtheory): for n from 2 to 800 do if ifactors(n)[2][nops(ifactors(n)[2])][1]! mod n <> 0 then printf(`%d, `, n) fi; od:
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MATHEMATICA
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Select[Range[330], Mod[FactorInteger[#][[-1, 1]]!, #] != 0 &]
(* From Jean-François Alcover, May 19 2011 *)
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PROG
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(PARI) is(n)=my(s=factor(n)[, 1]); s[#s]!%n>0 \\ Charles R Greathouse IV, Sep 20 2012
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CROSSREFS
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Subsequence of A122145.
Cf. A002034, A006530, A057108.
Sequence in context: A053443 A048098 A122145 * A069189 A069168 A102211
Adjacent sequences: A057106 A057107 A057108 * A057110 A057111 A057112
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, Aug 08 2000
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EXTENSIONS
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More terms from James A. Sellers, Aug 22 2000
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STATUS
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approved
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