A187142 Smallest number k such that the continued fraction expansion of sqrt(k) contains n distinct numbers.

1, 2, 7, 14, 19, 61, 94, 151, 211, 436, 604, 844, 919, 1324, 1894, 2011, 2731, 3691, 4951, 5086, 6451, 7606, 9619, 10294, 13126, 15814, 17599, 21499, 19231, 21319, 30319, 31606, 34654, 42379, 46006, 53299, 48799, 60811, 76651, 78094, 85999, 90931

Offset

1,2

Comments

For the first 191 terms, a(n) has the form p*2^i, where p is prime and i >= 0. - T. D. Noe, Mar 07 2011

Looking at just the periodic part of sqrt(k), it is the same sequence without the term a(1). - Robert G. Wilson v, Mar 22 2011

Conjecture: a(n) is of the form p, 2*p or 4*p, where p is prime. For the first 528 terms, a(n) is of the form 4*p only for n = 10, 11, 12, 14, 81 and 277. - Chai Wah Wu, Oct 04 2019

Links

Chai Wah Wu, Table of n, a(n) for n = 1..528 (terms 1..191 from Robert G. Wilson v)

Example

ContinuedFraction(sqrt(2),x) => 1,2,2,2,...: two distinct terms (1,2);
sqrt(7) => 2,1,1,1,4,1,1,1,...: three distinct terms (1,2,4);
sqrt(14) => four distinct terms (1,2,3,6);
sqrt(19) => five distinct terms (1,2,3,4,8).

Mathematica

f[n_] := Length@ Union@ Flatten@ ContinuedFraction@ Sqrt@ n; t = Table[ 0, {100}]; Do[a = f@ k; If[ a <= 100 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]], {k, 10^5}]; t

Crossrefs

Cf. A040000, A010121.

Sequence in context: A228831 A285682 A018363 * A263398 A161702 A114346

Adjacent sequences: A187139 A187140 A187141 * A187143 A187144 A187145

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Status

approved