OFFSET
1,1
COMMENTS
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 (Erdős 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
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284-292.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..50000 (first 1750 terms from Vincenzo Librandi)
Paul Erdős and Ilias Kastanas, Solution 6674: The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
Steven R. Finch, The Average Value of the Smarandache Function [Broken link]
Steven R. Finch, The Average Value of the Smarandache Function
Kevin Ford, On integers n for which n does not divide P(n)!, University of Illinois at Urbana-Champaign (2019).
A. Ivic (2004), On a problem of Erdos involving the largest prime factor of n, arXiv:math/0311056 [math.NT], 2003-2004.
C. Rivera, Conjecture about their density
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function
EXAMPLE
12 is in the sequence since 3 is the largest prime factor of 12, but 12 is not a factor of 3! = 6.
MAPLE
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:
MATHEMATICA
Select[Range[330], Mod[FactorInteger[#][[-1, 1]]!, #] != 0 &] (* Jean-François Alcover, May 19 2011 *)
PROG
(PARI) is(n)=my(s=factor(n)[, 1]); s[#s]!%n>0 \\ Charles R Greathouse IV, Sep 20 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Aug 08 2000
EXTENSIONS
More terms from James A. Sellers, Aug 22 2000
STATUS
approved