A307453 a(n) is the least prime p for which the continued fraction expansion of sqrt(p) has exactly n consecutive 1's starting at position 2.

2, 3, 31, 7, 13, 3797, 5273, 4987, 90371, 79873, 2081, 111301, 1258027, 5325101, 12564317, 9477889, 47370431, 709669249, 1529640443, 2196104969, 392143681, 8216809361, 30739072339, 200758317433, 370949963971, 161356959383, 1788677860531, 7049166342469, 4484287435283, 3690992602753

Offset

0,1

Links

Table of n, a(n) for n=0..29.

Piotr Miska, Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, arXiv:1904.03404 [math.NT], 2019. See Table 1 p. 15.

Index entries for continued fractions for constants

Formula

Limit_{n->infinity} (sqrt(a(n)) - floor(sqrt(a(n)))) = A094214. - Daniel Suteu, Apr 09 2019

Example

For p = 2,  we have [1; 2, ...]; see A040000.
For p = 3,  we have [1; 1, 2, ...]; see A040001.
For p = 31, we have [5; 1, 1, 3, ...]; see A010129.
For p = 7,  we have [2; 1, 1, 1, 4, ...]; see A010121.

Prog

(PARI) isok(p, n) = {my(c=contfrac(sqrt(p)));  for (k=2, n+1, if (c[k] != 1, return (0)); ); return(c[n+2] !=  1); }
a(n) = {my(p=2); while (! isok(p, n), p = nextprime(p+1)); p; }

Crossrefs

Cf. A040000, A040001, A010121, A010129.

Sequence in context: A093712 A035514 A114009 * A143665 A074479 A272043

Adjacent sequences: A307450 A307451 A307452 * A307454 A307455 A307456

Keyword

nonn

Author

Michel Marcus, Apr 09 2019

Extensions

a(21)-a(29) from Daniel Suteu, Apr 09 2019

a(0) added by Chai Wah Wu, Apr 09 2019

Status

approved