cp's OEIS Frontend

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

%I A057109 #52 Jul 02 2025 16:02:00
%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 that are not factors of P(n)!, where P(n) is the largest prime factor of n.
%C A057109 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
%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 (Erdős and Kastanas 1994, Ivic 2004). - _Jonathan Sondow_, Jan 10 2005
%C A057109 Same as numbers n such that |e - m/n| < 1/(P(n)+1)! for some integer m. - _Jonathan Sondow_, Dec 29 2007
%D A057109 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284-292.
%H A057109 Alois P. Heinz, <a href="/A057109/b057109.txt">Table of n, a(n) for n = 1..50000</a> (first 1750 terms from Vincenzo Librandi)
%H A057109 Paul Erdős and Ilias Kastanas, <a href="http://www.jstor.org/stable/2324376">Solution 6674: The smallest factorial that is a multiple of n</a>, Amer. Math. Monthly 101 (1994) 179.
%H A057109 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/golomb/smarand/smarand1.html">The Average Value of the Smarandache Function</a> [Broken link]
%H A057109 Steven R. Finch, <a href="http://fs.unm.edu/SF/TheAverageValue.pdf">The Average Value of the Smarandache Function</a>
%H A057109 Kevin Ford, <a href="https://faculty.math.illinois.edu/~ford/wwwpapers/nPn.pdf">On integers n for which n does not divide P(n)!</a>, University of Illinois at Urbana-Champaign (2019).
%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>, arXiv:math/0311056 [math.NT], 2003-2004.
%H A057109 C. Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_027.htm">Conjecture about their density</a>
%H A057109 J. Sondow, <a href="http://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%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>, arXiv:0704.1282 [math.HO], 2007-2010.
%H A057109 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</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:
%t A057109 Select[Range[330],Mod[FactorInteger[#][[-1,1]]!,#] != 0 &] (* _Jean-François Alcover_, May 19 2011 *)
%o A057109 (PARI) is(n)=my(s=factor(n)[, 1]); s[#s]!%n>0 \\ _Charles R Greathouse IV_, Sep 20 2012
%Y A057109 Subsequence of A122145.
%Y A057109 Cf. A002034, A006530, A057108.
%K A057109 easy,nonn
%O A057109 1,1
%A A057109 _Henry Bottomley_, Aug 08 2000
%E A057109 More terms from _James Sellers_, Aug 22 2000