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.

1, 5, 15, 507

Offset: 1

Author

Adam Vulic, Apr 11 2024

Keywords

• nonn
• base
• more

Comments

From David A. Corneth, Apr 11 2024: (Start)

This sequence is well defined as A030000 is well defined; every finite string of digits is contained in some power of 2.

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)

Examples

			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.)
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.)
		

Links

• Brady Haran, Apocalyptic Numbers, Numberphile video, 2024.

Crossrefs

Cf. A018856, A005054, A030000, A034887.

Programs

Mathematica

k = 0; Rest@ NestList[(While[SequenceCount[IntegerDigits[2^k], IntegerDigits[2^#]] == 0, k++]; k++; k - 1) &, 1, 4] (* Michael De Vlieger, Apr 19 2024 *)