A233593 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/....)) is aperiodic.

7, 13, 14, 19, 21, 22, 23, 28, 29, 31, 32, 33, 34, 41, 43, 46, 47, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 85, 86, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 103, 106, 107

Offset: 1

Author

Stanislav Sykora, Jan 06 2014

Keywords

• nonn

Comments

For more details about this type of expansions, see A233582.

The cases with known periodic expansions, listed in A233592, all become periodic after just two leading terms. In contrast, the Blazys's expansion of sqrt(a(k)) for every member a(k) of this list remains aperiodic up to at least 1000 terms. It is therefore conjectured, though not proved, that these expansions are indeed aperiodic.

Examples

			Blazys's expansion of sqrt(7), A233587, is {2, 3, 30, 34, 111, ...}. Its first 1000 terms are all distinct. Hence, 7 is a term of this sequence.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..200
• Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
• Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki.

Crossrefs

Cf. A233592.

Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.