A031749 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 71.

5043, 20168, 45375, 80664, 126035, 181488, 247023, 322640, 408339, 504120, 609983, 725928, 851955, 988064, 1134255, 1290528, 1456883, 1633320, 1819839, 2016440, 2223123, 2439888, 2666735, 2903664, 3150675, 3407768, 3674943, 3952200, 4239539

Offset

1,1

Comments

The continued fraction expansion of sqrt((j*m)^2+t*m) for m >= 1 where t divides 2*j has the form [j*m, 2*j/t, 2*j*m, 2*j/t, 2*j*m, ...]. Thus numbers of the form (71*m)^2 + 2*m for m >= 1 are in the sequence. Are there any others? - Chai Wah Wu, Jun 18 2016

The term 25776072 is not of the form (71*m)^2 + 2*m. - Chai Wah Wu, Jun 19 2016

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Mathematica

lt71Q[n_]:=Module[{s=Sqrt[n]}, If[IntegerQ[s], 0, Min[ContinuedFraction[s] [[2]]]] == 71]; Select[Range[43*10^5], lt71Q] (* Harvey P. Dale, Apr 11 2017 *)

Prog

(Python)
from sympy import continued_fraction_periodic
A031749_list = [n for n, d in ((n, continued_fraction_periodic(0, 1, n)[-1]) for n in range(1, 10**5)) if isinstance(d, list) and min(d) == 71] # Chai Wah Wu, Jun 09 2017

Crossrefs

Cf. A031424.

Sequence in context: A229674 A243133 A031569 * A229857 A155145 A281679

Adjacent sequences: A031746 A031747 A031748 * A031750 A031751 A031752

Keyword

nonn

Author

David W. Wilson

Status

approved