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EXAMPLE
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sqrt(prime(97)) = sqrt(509) has continued fraction [22; 1, 1, 3, 1, 1, 2, 10, 1, 8, 8, 1, 10, 2, 1, 1, 3, 1, 1, 44, ...], period 19.
sqrt(prime(98)) = sqrt(521) has continued fraction [22; 1, 4, 1, 2, 1, 2, 8, 1, 3, 3, 1, 8, 2, 1, 2, 1, 4, 1, 44, ...], period 19.
These are the first 2 consecutive primes with the same period of continued fraction for square root, so a(2) = 97.
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MATHEMATICA
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A054269[n_] := Module[{s = Sqrt[Prime[n]]}, If[IntegerQ[s], 0, Length[ContinuedFraction[s][[2]]]]]; Do[find = 0; k = 0; While[find == 0, k++; If[Length[Union[Table[A054269[j], {j, k, k + n - 1}]]] == 1, find = 1; Print[k]]], {n, 1, 3}]
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