User blog comment:Ikosarakt1/Apocalyptic function/@comment-5529393-20130321020848

\(2^n\) has about \(n log 2\) digits, where log is the base 10 logarithm. The probability that a 3-digit string is 666 is .001, so the probability that a number of the form \(2^n\) is not apocalyptic is

(.999)^{n log 2} = ((.999)^{log 2})^n ~ (.999698865)^n

Summing over all n, we get

Sum_n (.999698865)^n = 1/(1 - .999698865) ~ 3320.767

So there are probably about 3300-3340 nonapocalyptic numbers. I don't know about how many gaps there will be, but obviously there would be fewer. So Ap(n) with n > 3340 is likely undefined.