[seqfan] A265739, A265742, nice idea, but . . .

[Brad Klee]

Both /pi/ and /e/ are known irrational numbers. There must be an infinite
set of rationals satisfying Dirichlet's irrationality criterion. These
sequences A265739 and A265742 seem to record denominators of just those
numbers. Compare with A002486 and A007677, with partial overlap. Aren't the
conjectures superfluous?

While we are on the topic, it's worth mentioning a couple of good reads,
which may lead to fun experimentation with OEIS: "A Rational Approach to
Pi" [1] and "Searching for Apéry-Style Miracles" [2]. Beuker's caveat
emptor "long run" (cf. A123178) should be taken seriously. I'm not sure of
the next integer after n=3 where the integral approximation /K_i/ visits a
preferable approximant? If similar deception occurs with Apéry-style
approximations, this would make the question of geussing between major and
minor miracles more difficult during the "reconnaissance" (ha!) phase.

The OEIS already contains a number of second order non-linear recurrences,
so we could begin the task of identifying integer sequences with Minor /
Major approximation Miracles. For example, I recently reconnoitred that
both A295870 and A005721 adhere to recurrences associated with /pi/
miracles. Unfortunately I have yet to think of good methods to prove
convergence properties, while standard proof techniques are slow to trickle
down.

Under another flag,

Brad

[1] http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf
[2] https://arxiv.org/abs/1405.4445