A102581 Numbers n such that denominator of Sum_{k=0 to 2n+1} 1/k! is (2n+1)!/2.

1, 2, 6, 10, 30, 32, 42, 46, 56, 62, 70, 80, 82, 96, 120, 122, 136, 150, 160, 162, 170, 172, 176, 186, 192, 196, 200, 210, 222, 230, 236, 252, 262, 266, 276, 290, 292, 300, 302, 306, 312, 326, 356, 366, 380, 382, 400, 416, 422, 426, 452, 460, 486, 490, 496, 500

Offset

1,2

Comments

The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m > 1 is odd, say m = 2n+1, then d is even. n is a member when d = 2. If m > 3 and m = 3 (mod 4), so that n > 1 is odd, then d is divisible by 4. So except for 1 the members are even.

Links

Table of n, a(n) for n=1..56.

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

Formula

a(n) = 2*A102582(n-1) for n > 1.

Example

1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 = (2*1+1)!/2, so 1 is a member.

Mathematica

fQ[n_] := (Denominator[ Sum[1/k!, {k, 0, 2n + 1}]] == (2n + 1)!/2); Select[ Range[ 500], fQ[ # ] &] (* Robert G. Wilson v, Jan 24 2005 *)

Crossrefs

n is a member <=> A093101(2n+1) = 2 <=> A061355(2n+1) = (2n+1)!/2 <=> n = 1 or n/2 is a member of A102582.

Sequence in context: A364879 A192616 A243393 * A152000 A038042 A032374

Adjacent sequences: A102578 A102579 A102580 * A102582 A102583 A102584

Keyword

nonn

Author

Jonathan Sondow, Jan 21 2005

Extensions

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

Status

approved