A187261 Least number k such that the continued fraction expansion of its square root contains the first n natural numbers.

1, 2, 14, 19, 211, 463, 634, 1057, 1951, 2326, 4156, 5149, 8254, 9811, 10651, 21319, 15814, 19609, 29527, 42379, 46006, 58171, 89959, 97579, 144271, 135319, 164431, 217519, 201919, 230101, 216451, 285814, 307759, 323359

Offset: 1

Author

Robert G. Wilson v, Mar 07 2011

Keywords

• nonn

Examples

			a(1) is 1.
a(2) is 2 because the cf of sqrt(2) = 1, 2, 2, .., .
a(3) is 14 because the cf of sqrt(14) = 3, {1, 2, 1, 6},
a(4) is 19 because the cf of sqrt(19) = 4, {2, 1, 3, 1, 2, 8},
a(5) is 211 because the cf of sqrt(211) = 14, {1, 1, 9, 5, 1, 2, 2, 1, 1, 4, 3, 1, 13, 1, 3, 4, 1, 1, 2, 2, 1, 5, 9, 1, 1, 28} which includes the natural numbers 1 through 5 and there does not exist any integer less than 211 which has this characteristics, etc.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000027, A187142.

Programs

Mathematica

f[n_] := Block[{cf = Union@ Flatten@ ContinuedFraction@ Sqrt@ n, k = 1}, While[k <= Length[cf] && k == cf[[k]], k++]; k - 1]; t = Table[ 0, {100}]; k = 1; While[k < 10^7, a = f@ k; If[a <= Length[t] && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t