A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 175, 176, 195, 208, 224, 255, 288, 323, 360, 399, 440, 483, 528, 551, 575, 624, 675, 728, 783, 799, 840, 899, 960, 1023, 1035, 1088, 1155, 1224, 1247, 1295, 1368, 1403, 1443, 1520, 1599, 1680, 1763, 1848, 1872

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

All the terms of A005563 are here, as well as some additional terms (with even period > 2 and the digit 1 in central position) (e.g., sqrt(175) = [13,'4, 2, 1, 2, 4, 26']).

References

• Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Robert Macmillan, Continued fractions, Math. Gaz., Vol. 84, No. 499 (2000), pp. 30-35. See p. 34.

Crossrefs

Union of A005563 and A102538.

Cf. A040001, A040005, A040011, A040019, A040029, A062196.

Programs

Mathematica

Select[ Range@ 1900, !IntegerQ[ Sqrt@ #] && Count[ ContinuedFraction[ Sqrt@ #][[2]], 1] == 1 &] (* Robert G. Wilson v, Jul 03 2011 *)

Extensions

Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001

Incorrect formulas and programs removed by R. J. Mathar, Jan 06 2011