This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371887 #27 May 11 2024 22:00:23 %S A371887 1,5,15,507 %N A371887 a(1) = 1; for n > 1, a(n) is the smallest positive integer k such that the digits of 2^k contain 2^a(n-1) as a proper substring. %C A371887 From _David A. Corneth_, Apr 11 2024: (Start) %C A371887 This sequence is well defined as A030000 is well defined; every finite string of digits is contained in some power of 2. %C A371887 An upper bound for a(n), n > 1, can be found by solving 2^k == 2^a(n-1) (mod 10^m) where m is the number of digits of 2^a(n-1) (cf. A034887). This gives a(n) <= k = a(n-1) + 4*5^(m-1) (cf. A005054). So a(5) <= 507 + 4*5^152, which is about 7*10^106. (End) %H A371887 Brady Haran, <a href="https://www.youtube.com/watch?v=0LkBwCSMsX4">Apocalyptic Numbers</a>, Numberphile video, 2024. %e A371887 a(2) is the smallest k > 0 such that the digits of 2^k contain 2^a(1) = 2^1 = 2 as a proper substring, so a(2) = 5. (2^5 = 32.) %e A371887 a(3) is the smallest k > 0 such that the digits of 2^k contain 2^a(2) = 32 as a proper substring, so a(3) = 15. (2^15 = 32768.) %t A371887 k = 0; Rest@ NestList[(While[SequenceCount[IntegerDigits[2^k], IntegerDigits[2^#]] == 0, k++]; k++; k - 1) &, 1, 4] (* _Michael De Vlieger_, Apr 19 2024 *) %Y A371887 Cf. A018856, A005054, A030000, A034887. %K A371887 nonn,base,more %O A371887 1,2 %A A371887 _Adam Vulic_, Apr 11 2024