

A057109


Numbers n which are not a factor of P(n)!, where P(n) is the largest prime factor of n.


11



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 (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


REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284292.
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) 637641.


LINKS

Table of n, a(n) for n=1..58.
S. R. Finch, The Average Value of the Smarandache Function
A. Ivic (2004), On a problem of Erdos involving the largest prime factor of n
C. Rivera, Conjecture about their density
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality
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 &] (* JeanFrançois 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

Subsequence of A122145.
Cf. A002034, A006530, A057108.
Sequence in context: A053443 A048098 A122145 * A069189 A069168 A102211
Adjacent sequences: A057106 A057107 A057108 * A057110 A057111 A057112


KEYWORD

easy,nonn


AUTHOR

Henry Bottomley, Aug 08 2000


EXTENSIONS

More terms from James A. Sellers, Aug 22 2000


STATUS

approved



