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A122378 Numbers m such that m^2 > S(m)!, where S(m)! is the smallest factorial divisible by m.

%I A122378 #26 Feb 16 2025 08:33:02
%S A122378 2,3,6,8,12,15,20,24,30,36,40,45,48,60,72,80,84,90,105,112,120,126,
%T A122378 140,144,168,180,210,224,240,252,280,288,315,320,336,360,384,420,448,
%U A122378 480,504,560,576,630,640,648,672,720,756,810,840,864,896,945,960,1008,1080
%N A122378 Numbers m such that m^2 > S(m)!, where S(m)! is the smallest factorial divisible by m.
%C A122378 It is conjectured that m^2 < S(m)! for almost all m.
%C A122378 For each k > 1, at most tau(k!)/2 = A000005(k!)/2 are in the sequence because of that k. So at most Sum_{k = 1..m} tau(k!)/(2*m!) of the numbers up to m! are terms. This tends to 0 as m tends to infinity. - _David A. Corneth_, Dec 29 2019
%H A122378 David A. Corneth, <a href="/A122378/b122378.txt">Table of n, a(n) for n = 1..10000</a>
%H A122378 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 A122378 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 A122378 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%H A122378 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%e A122378 15^2 = 225 > 120 = 5! = S(15)!, so 15 is a member.
%t A122378 nmax = 1100;
%t A122378 Do[m = 1; While[!IntegerQ[m!/n], m++]; S[n] = m, {n, 1, nmax}];
%t A122378 Select[Range[nmax], #^2 > S[#]!&] (* _Jean-François Alcover_, Dec 04 2018 *)
%o A122378 (PARI) upto(n) = {my(res = List(), maxf = 1, olddiv, newdiv, n2 = n^2, cf = 1); while(maxf! < n2, maxf++); maxf--; olddiv = divisors(0!); newdiv = divisors(1!); for(i = 2, maxf, olddiv = newdiv; cf*=i; newdiv = divisors(cf); cans = setminus(Set(newdiv), Set(olddiv)); for(j = 1, #cans, if(cans[j]^2 > cf, if(cans[j] <= n, listput(res, cans[j]) , next(2) ); ) ) ); listsort(res); res } \\ _David A. Corneth_, Dec 29 2019
%Y A122378 Cf. A000290, A002034, A057109, A092495, A122379, A122380.
%K A122378 nonn
%O A122378 1,1
%A A122378 _Jonathan Sondow_, Sep 03 2006