A338779 a(n) is the smallest number k such that period of continued fraction for sqrt(prime(j)) equal for all prime(k) <= prime(j) < prime(k + n).

1, 97, 141043

Offset: 1

Author

Ilya Gutkovskiy, Nov 08 2020

Keywords

• nonn
• bref
• more

Comments

The corresponding primes are 2, 509, 1885717, ...

Examples

			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.
		

Crossrefs

Cf. A003285, A031400, A054269.

Programs

Mathematica

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}]