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

2, 3, 6, 8, 12, 15, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 224, 240, 252, 280, 288, 315, 320, 336, 360, 384, 420, 448, 480, 504, 560, 576, 630, 640, 648, 672, 720, 756, 810, 840, 864, 896, 945, 960, 1008, 1080

Offset: 1

Author

Jonathan Sondow, Sep 03 2006

Keywords

• nonn

Comments

It is conjectured that m^2 < S(m)! for almost all m.

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

Examples

			15^2 = 225 > 120 = 5! = S(15)!, so 15 is a member.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• 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
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A000290, A002034, A057109, A092495, A122379, A122380.

Programs

Mathematica

nmax = 1100;
Do[m = 1; While[!IntegerQ[m!/n], m++]; S[n] = m, {n, 1, nmax}];
Select[Range[nmax], #^2 > S[#]!&] (* Jean-François Alcover, Dec 04 2018 *)

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