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A102581
Numbers n such that denominator of Sum_{k=0 to 2n+1} 1/k! is (2n+1)!/2.
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
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 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.
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
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 21 2005
EXTENSIONS
More terms from Robert G. Wilson v, Jan 24 2005
STATUS
approved