login
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.
3
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, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
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
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Jan 06 2014
STATUS
approved