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A129924 Primes p such that p divides both A061354(p-3) and A061354(p-1). 4
5, 13, 37, 463 (list; graph; refs; listen; history; text; internal format)
Conjecture: a(n) = A064384(n+1).
Also primes p such that p divides A120265(p-2), where A120265(n) = A061354(n) - A061355(n) = Numerator of Sum[1/k!,{k,1,n}].
The conjecture is true. It is the case n = p-3 of the relation GCD(A061354(n), A061354(n+2)) = A124779(n), which follows from the Comments in A064384 and A124779. For a proof, see the link "The Taylor series for e ...". - Jonathan Sondow, Jun 12 2007
Michael Mossinghoff has calculated that 5, 13, 37, 463 are the only terms up to 150 million. Heuristics suggest the sequence is infinite but very sparse. - Jonathan Sondow, Jun 12 2007
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.
g=1; Do[ g=g+1/n!; f=Numerator[g]; If[ PrimeQ[n+3] && IntegerQ[f/(n+3)], Print[n+3]], {n, 1, 1000}]
Cf. A061354 = Numerator of Sum_{k=0..n} 1/k!. Cf. A064384, A124779.
Cf. A120265 = Numerator of Sum[1/k!, {k, 1, n}]. Cf. A061355.
Sequence in context: A146452 A146062 A201612 * A360797 A080143 A077919
Alexander Adamchuk, Jun 06 2007

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