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A122379 Numbers n such that S(n)! > n^2 > P(n)!, where S(n)! is the smallest factorial divisible by n and P(n) is the greatest prime factor of n.

%I A122379 #24 Dec 30 2019 19:37:48
%S A122379 4,9,16,18,25,27,32,50,54,64,75,81,96,98,100,108,125,128,135,147,150,
%T A122379 160,162,175,189,192,196,200,216,225,243,245,250,256,270,294,300,324,
%U A122379 343,350,375,378,392,400,405,432,441,450,486,490,500,512,525,540,567
%N A122379 Numbers n such that S(n)! > n^2 > P(n)!, where S(n)! is the smallest factorial divisible by n and P(n) is the greatest prime factor of n.
%C A122379 It is conjectured that n^2 < P(n)! for almost all n. It is known that S(n) = P(n) for almost all n. Clearly, S(n) >= P(n) for all n > 1.
%H A122379 David A. Corneth, <a href="/A122379/b122379.txt">Table of n, a(n) for n = 1..10000</a>
%H A122379 J. Sondow, <a href="https://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 A122379 J. Sondow, <a href="https://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 A122379 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%e A122379 S(9)! = 6! = 720 > 81 = 9^2 > 6 = 3! = P(9)!, so 9 is a member.
%t A122379 s[n_] := For[k = 1, True, k++, If[Divisible[k!, n], Return[k]]];
%t A122379 p[n_] := FactorInteger[n][[-1, 1]];
%t A122379 okQ[n_] := s[n]! > n^2 > p[n]!;
%t A122379 Select[Range[1000], okQ] (* _Jean-François Alcover_, Jan 27 2019 *)
%Y A122379 Cf. A000290, A002034, A006530, A057109, A092495, A102068, A122378, A122380.
%K A122379 nonn
%O A122379 1,1
%A A122379 _Jonathan Sondow_, Sep 03 2006