A057109 Numbers n that are not factors of P(n)!, where P(n) is the largest prime factor of n.

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

Offset: 1

Author

Henry Bottomley, Aug 08 2000

Keywords

• easy
• nonn

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

Examples

			12 is in the sequence since 3 is the largest prime factor of 12, but 12 is not a factor of 3! = 6.
		

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

Crossrefs

Subsequence of A122145.

Cf. A002034, A006530, A057108.

Programs

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 *)

PARI

is(n)=my(s=factor(n)[, 1]); s[#s]!%n>0 \\ Charles R Greathouse IV, Sep 20 2012

Extensions

More terms from James Sellers, Aug 22 2000