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

1,1

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.

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.

Example

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.

Crossrefs

Cf. A233592.

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

Sequence in context: A342581 A353443 A332480 * A013651 A050931 A072864

Adjacent sequences: A233590 A233591 A233592 * A233594 A233595 A233596

Keyword

nonn

Author

Stanislav Sykora, Jan 06 2014

Status

approved