A102582 Numbers n such that denominator of Sum_{k=0..4n+1} 1/k! is (4n+1)!/2.

1, 3, 5, 15, 16, 21, 23, 28, 31, 35, 40, 41, 48, 60, 61, 68, 75, 80, 81, 85, 86, 88, 93, 96, 98, 100, 105, 111, 115, 118, 126, 131, 133, 138, 145, 146, 150, 151, 153, 156, 163, 178, 183, 190, 191, 200, 208, 211, 213, 226, 230, 243, 245, 248, 250, 256, 260, 261, 265

Offset: 1

Author

Jonathan Sondow, Jan 21 2005

Keywords

• nonn

Comments

The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m = 4n+1 > 1, then d is even. n is a member when d = 2.

Examples

			1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! = 163/60 and 60 = 5!/2 = (4*1+1)!/2, so 1 is 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

n is a member <=> 2n is a member of A102581 <=> A093101(4n+1) = 2 <=> A061355(4n+1) = (4n+1)!/2.

Programs

Mathematica

fQ[n_] := (Denominator[ Sum[1/k!, {k, 0, 4n + 1}]] == (4n + 1)!/2); Select[ Range[0, 274], fQ[ # ] &] (* Robert G. Wilson v, Jan 24 2005 *)

Formula

a(n) = A102581(n+1)/2.

Extensions

More terms from Robert G. Wilson v, Jan 24 2005