%I A057109
%S A057109 4,8,9,12,16,18,24,25,27,32,36,45,48,49,50,54,64,72,75,80,81,90,96,98,
%T A057109 100,108,121,125,128,135,144,147,150,160,162,169,175,180,189,192,196,
%U A057109 200,216,224,225,240,242,243,245,250,256,270,288,289,294,300,320,324
%N A057109 Numbers n which are not a factor of P(n)!, where P(n) is the largest 
               prime factor of n.
%C A057109 These are also the numbers whose Smarandache function is composite. Their 
               density approaches zero as they go to infinity. - Jud McCranie (j.mccranie(AT)comcast.net), 
               Dec 08 2001
%C A057109 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 (jsondow(AT)alumni.princeton.edu), 
               Jan 10 2005
%C A057109 Same as numbers n such that |e - m/n| < 1/(P(n)+1)! for some integer 
               m. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 29 2007
%D A057109 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284-292.
%D A057109 P. Erdos and I. Kastanas, Problem/Solution 6674:The smallest factorial 
               that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
%D A057109 J. Sondow, A geometric proof that e is irrational and a new measure of 
               its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
%H A057109 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/golomb/smarand/
               smarand1.html">The Average Value of the Smarandache Function</a>
%H A057109 C. Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_027.htm">
               Conjecture about their density</a>
%H A057109 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmarandacheFunction.html">Smarandache function</a>
%H A057109 A. Ivic (2004), <a href="http://arXiv.org/abs/math/0311056">On a problem 
               of Erdos involving the largest prime factor of n</a>
%H A057109 J. Sondow, <a href="http://arXiv.org/abs/0704.1282">A geometric proof 
               that e is irrational and a new measure of its irrationality</a>
%e A057109 12 is in the sequence since 3 is the largest prime factor of 12, but 
               12 is not a factor of 3!=6.
%p A057109 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:
%Y A057109 Cf. A002034, A006530, A057108.
%Y A057109 Sequence in context: A053443 A048098 A122145 this_sequence A069189 A069168 
               A102211
%Y A057109 Adjacent sequences: A057106 A057107 A057108 this_sequence A057110 A057111 
               A057112
%K A057109 easy,nonn
%O A057109 1,1
%A A057109 Henry Bottomley (se16(AT)btinternet.com), Aug 08 2000
%E A057109 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000