login
Numbers from an irrationality measure for e, with a(1) = 2.
4

%I #20 Dec 04 2018 12:04:24

%S 2,3,4,5,6,4,8,5,7,6,12,5,14,8,6,7,18,7,20,6,8,12,24,5,11,14,10,8,30,

%T 6,32,9,12,18,8,7,38,20,14,6,42,8,44,12,7,24,48,7,15,11,18,14,54,10,

%U 12,8,20,30,60,6,62,32,8,9,14,12,68,18,24,8,72,7,74,38,11,20,12,14,80,7,10

%N Numbers from an irrationality measure for e, with a(1) = 2.

%C If n > 1, then a(n)! is the smallest factorial such that |e - m/n| > 1/a(n)! for any integer m.

%C a(n)! is the second smallest factorial divisible by n.

%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/21-24-2012/azarianIJCMS21-24-2012.pdf">Euler's Number Via Difference Equations</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.

%H 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 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 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>

%F a(n) = A002034(n) + 1.

%e a(6) = S(6) + 1 = 3 + 1 = 4.

%t nmax = 100; Do[m=1; While[!IntegerQ[m!/n], m++]; a[n] = m+1, {n, 1, nmax}];

%t Array[a, nmax] (* _Jean-François Alcover_, Dec 04 2018 *)

%Y Cf. A001113, A002034, A092495, A122417.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Sep 03 2006