Define apocalyptic gap as to be gap in the sequence of apocalyptic numbers that filled with non-apocalyptic integer powers of two. The first gap starts with \(2^0 = 1\) and ends with \(2^{156}\). The second gap starts with \(2^{158}\) and ends with \(2^{191}\). In the expression \(2^n\), as n grows, chances of \(2^n\) to be apocalyptic increases. Therefore, gaps appears rarer and rarer, and I can define the function Ap(n) to be the largest number in the n-th gap. The first few terms are easy to compute:

  • \(Ap(1) = 2^{156}\)
  • \(Ap(2) = 2^{191}\)
  • \(Ap(3) = 2^{217}\)
  • \(Ap(4) = 2^{219}\)
  • \(Ap(5) = 2^{221}\)
  • \(Ap(6) = 2^{223}\)
  • \(Ap(7) = 2^{225}\)
  • \(Ap(8) = 2^{242}\)
  • \(Ap(9) = 2^{244}\)
  • \(Ap(10) = 2^{246}\)

So far it doesn't seem to grows faster, but, as I noticed above, the density of apocalyptic numbers becomes thicker and thicker, and computing Ap(1000000) or, say, Ap(1010100) is really hard, because digits of \(2^n\) changes irrationally, and finding the block "666" becomes the "brute-force" task. There is example of irrational, but computable function. However, what is the asymptotic growth of that function?

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