

A187142


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


2



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
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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



