A102582 -id:A102582 - cp's OEIS frontend

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

Jonathan Sondow, Jan 21 2005

nonn

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.

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

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

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

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

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

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

A102584 a(n) = 1/2 times the cancellation factor in reducing Sum_{k=0 to 2n+1} 1/k! to lowest terms.

Original entry on oeis.org

1, 1, 10, 5, 4, 1, 2, 65, 2000, 1, 26, 247, 20, 5, 2, 19, 8, 115, 10, 23, 52, 31, 10, 65, 416, 37, 2, 25, 20, 1, 38, 1, 40, 325, 1406, 37, 676, 65, 10, 63829, 368, 1, 230, 5, 4, 1, 26, 5, 40, 247, 26, 43, 3100, 9785, 2, 1, 256, 5, 2050, 13, 388, 1, 4810, 1495, 8, 23, 254, 5

Offset: 1

Jonathan Sondow, Jan 22 2005

nonn

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

			1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! = 13700/5040 = (20*685)/(20*252) and 7 = 2*3+1, so a(3) = 20/2 = 10.

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, arXiv:0709.0671 [math.NT], 2007-2009; 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.

a(n) = A093101(2n+1)/2 = (2n+1)!/(2*A061355(2n+1)).

cp's OEIS Frontend

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

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