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
Keywords
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
Programs
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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