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

1,1

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A131723 A198442 A035106 * A181687 A194881 A361922

Adjacent sequences: A122375 A122376 A122377 * A122379 A122380 A122381

Keyword

nonn

Author

Jonathan Sondow, Sep 03 2006

Status

approved