A377815 Lexicographically earliest infinite sequence of distinct positive integers such that the binary concatenation of its terms yields the same string as the binary concatenation of the binary weights of its terms.

1, 5, 2, 3, 4, 8, 15, 255, 7, 11, 13, 14, 16, 19, 21, 22, 6, 25, 32, 9, 26, 63, 65535, 23, 28, 10, 12, 64, 17, 95, 111, 128, 27, 256, 4294967295, 29, 35, 18, 20, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 24, 70, 73, 512, 30, 33, 31, 39, 18446744073709551615

Offset: 1

Author

Dominic McCarty, Nov 08 2024

Keywords

• nonn
• base

Comments

The sequence makes huge jumps. For example, here are three consecutive terms: a(70) = 88, a(71) = 2^256-1, a(72) = 97.

Runs of 0 bits induce large terms since z consecutive 0 bits becomes a term with weight at least 2^z and the smallest such is 2^(2^z) - 1.

The base-2 analog of A302656. The first b terms of this sequence's base-b analog are 1,2,...,(b-1), followed by (b^2+b-1).

Examples

			(a(n)):
1,   5,  2,  3,   4,    8,   15,      255,   7, ...
(a(n)) in binary:
1, 101, 10, 11, 100, 1000, 1111, 11111111, 111, ...
Binary weights of (a(n)):
1,   2,  1,  2,   1,    1,    4,        8,   3, ...
Binary weights of (a(n)) in binary:
1,  10,  1, 10,   1,    1,  100,     1000,  11, ...
The two binary lines are the same when concatenated.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..451
• Dominic McCarty, Log log scatterplot of (n, a(n)) for 1 < n <= 10000
• Dominic McCarty, Table of n, a(n), binary weight of a(n) for n = 1..100000

Crossrefs

Cf. A302656