A122417 Factorials from an irrationality measure for e, with a(1) = 2.

2, 6, 24, 120, 720, 24, 40320, 120, 5040, 720, 479001600, 120, 87178291200, 40320, 720, 5040, 6402373705728000, 5040, 2432902008176640000, 720, 40320, 479001600, 620448401733239439360000, 120, 39916800, 87178291200, 3628800, 40320

Offset: 1

Author

Jonathan Sondow, Sep 03 2006

Keywords

• nonn

Comments

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

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

Examples

			a(6) = (S(6)+1)! = (3+1)! = 24.
		

Links

• Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
• 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.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A001113, A002034, A092495, A122416.

Programs

Mathematica

nmax = 28;
Do[m = 1; While[!IntegerQ[m!/n], m++]; a[n] = (m+1)!, {n, 1, nmax}];
Array[a, nmax] (* Jean-François Alcover, Dec 04 2018 *)

Formula

a(n) = (A002034(n)+1)! = A122416(n)!.