1,1

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 (jsondow(AT)alumni.princeton.edu), 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 (jsondow(AT)alumni.princeton.edu), Jun 12 2007

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection

J. Sondow, Which Partial Sums of the Taylor Series for e Are Convergents to e? (and a Link to the Primes $2, 5, 13, 37, 463, ...$) with an Appendix "Periodic Behaviour of Some Recurrence Sequences Related to $e$, Modulo Powers of 2" by Kyle Schalm

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.

Adjacent sequences: A129921 A129922 A129923 this_sequence A129925 A129926 A129927

Sequence in context: A083413 A071100 A125734 this_sequence A080143 A077919 A026069

bref,hard,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 06 2007