A102470 Numbers n such that denominator of Sum_{k=0 to n} 1/k! is n!.

0, 1, 2, 4, 6, 8, 10, 16, 18, 20, 26, 28, 40, 46, 48, 58, 66, 68, 70, 80, 86, 96, 98, 118, 126, 130, 136, 146, 150, 170, 176, 178, 180, 188, 190, 206, 208, 210, 216, 230, 260, 266, 268, 278, 286, 288, 300, 306, 308, 326, 328, 338, 346, 358, 366, 370, 378, 380, 388

Offset: 1

Author

Jonathan Sondow, Jan 14 2005

Keywords

• nonn

Comments

a(n) is even for n > 1, as Sum_{k=0 to n} 1/k! reduces to lower terms when n > 1 is odd.

Examples

			1/0! + 1/1! + 1/2! + 1/3! +1/4! = 65/24 and 24 = 4!, so 4 is a member. But 1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 < 3!, so 3 is not a member.
		

Links

• 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 K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
• Index entries for sequences related to factorial numbers

Crossrefs

For n > 0, n is a member <=> A093101(n) = 1 <=> A061355(n) = n! <=> A061355(n) = A002034(A061355(n))! <=> A061354(n) = 1+n+n(n-1)+n(n-1)(n-2)+...+n!. See also A102471.

Programs

Mathematica

fQ[n_] := (Denominator[Sum[1/k!, {k, 0, n}]] == n!); Select[ Range[0, 389], fQ[ # ] &] (* Robert G. Wilson v, Jan 15 2005 *)

Formula

a(n) = 2*A102471(n-1) for n > 1.

Extensions

More terms from Robert G. Wilson v, Jan 15 2005