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

Author

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Keywords

• nonn

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A040000, A010121.

Programs

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