cp's OEIS Frontend

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

%I A187142 #38 Oct 20 2019 03:25:14
%S A187142 1,2,7,14,19,61,94,151,211,436,604,844,919,1324,1894,2011,2731,3691,
%T A187142 4951,5086,6451,7606,9619,10294,13126,15814,17599,21499,19231,21319,
%U A187142 30319,31606,34654,42379,46006,53299,48799,60811,76651,78094,85999,90931
%N A187142 Smallest number k such that the continued fraction expansion of sqrt(k) contains n distinct numbers.
%C A187142 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
%C A187142 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
%C A187142 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
%H A187142 Chai Wah Wu, <a href="/A187142/b187142.txt">Table of n, a(n) for n = 1..528</a> (terms 1..191 from Robert G. Wilson v)
%e A187142 ContinuedFraction(sqrt(2),x) => 1,2,2,2,...: two distinct terms (1,2);
%e A187142 sqrt(7) => 2,1,1,1,4,1,1,1,...: three distinct terms (1,2,4);
%e A187142 sqrt(14) => four distinct terms (1,2,3,6);
%e A187142 sqrt(19) => five distinct terms (1,2,3,4,8).
%t A187142 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
%Y A187142 Cf. A040000, A010121.
%K A187142 nonn
%O A187142 1,2
%A A187142 _Vladimir Joseph Stephan Orlovsky_, Mar 05 2011