A266568 a(n) = smallest k such that 2^k ends in a string of exactly n nonzero digits.

0, 4, 7, 13, 14, 18, 50, 24, 27, 31, 34, 37, 68, 93, 49, 51, 116, 214, 131, 155, 67, 72, 76, 77, 81, 86, 149, 498, 154, 286, 359, 866, 1225, 329, 664, 129, 573, 176, 655, 820, 571, 434, 1380, 475, 1260, 2251, 6015, 3066, 1738, 2136, 2297, 432, 665, 229, 1899

Offset: 1

Author

Jon E. Schoenfield, Jan 01 2016

Keywords

• nonn
• base

Comments

Since 2^a(n) must have at least n digits, a(n) >= (n-1)*log_2(10).

The 26-digit number 2^86 = 77371252455336267181195264 is almost certainly the largest power of 2 that contains no zero digit.

A notably low local minimum occurs at a(36) = 129, which is less than a(n) for all n > 26.

A notably high local maximum occurs at a(122) = 11267047.

Examples

			2^0 = 1 is the smallest power of 2 ending in a string ("1") of exactly 1 nonzero digit, so a(1) = 0.
2^4 = 16 is the smallest power of 2 ending in a string ("16") of exactly 2 nonzero digits, so a(2) = 4.
2^50 = 1125899906842624 is the smallest power of 2 ending in a string ("6842624") of exactly 7 nonzero digits, so a(7) = 50.
The last 7 digits of 2^24 = 16777216 -- i.e., "6777216" -- are also nonzero, but so is the preceding digit, so 2^24 ends in a string of exactly 8 nonzero digits. Since no smaller power of 2 ends in exactly 8 nonzero digits, a(8) = 24.
		

Crossrefs

Cf. A007377, A031140, A031141, A031142, A031143, A181611.