A304588 Length of shortest prefix of the Thue-Morse word (A010060) such that some length-n block appears twice.

3, 5, 9, 10, 17, 18, 19, 20, 33, 34, 35, 36, 37, 38, 39, 40, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266

Offset: 1

Author

Jeffrey Shallit, May 15 2018

Keywords

• nonn

Comments

lim inf a(n)/n = 5/2 and lim sup a(n)/n = 4.

a(n) is "2-sychronized", which means that there is an automaton that accepts, in parallel, the base-2 expansions of n and a(n). For this sequence an 8-state automaton suffices. - Jeffrey Shallit, Mar 06 2020

Examples

			For n = 3 we have a(3) = 9 because the first 9 symbols of Thue-Morse are 011010011, and 011 is the first length-3 prefix to be repeated in this prefix.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..4096
• Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Arxiv preprint arXiv:1510.00282 [math.NT], October 1 2015.
• Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 931-947.
• Rémy Sigrist, PARI program for A304588

Crossrefs

Cf. A010060, A004756.

Programs

PARI

See Links section.

Formula

Apparently, a(n+1) = A004756(n) + 1. - Rémy Sigrist, Nov 04 2020