A035015 Period of continued fraction for square root of n-th squarefree integer.

1, 2, 1, 2, 4, 1, 2, 5, 4, 2, 1, 6, 6, 6, 4, 1, 5, 2, 8, 4, 4, 2, 1, 2, 2, 3, 2, 10, 12, 4, 2, 5, 4, 6, 7, 6, 11, 4, 1, 2, 10, 8, 6, 8, 7, 5, 6, 4, 4, 1, 2, 5, 10, 2, 5, 8, 10, 16, 4, 11, 1, 2, 12, 2, 9, 6, 15, 2, 6, 9, 6, 10, 10, 4, 1, 2, 12, 10, 3, 6, 16, 14, 9, 4, 18, 4, 4, 2, 1, 2, 9, 20, 10, 4

Offset

2,2

Comments

Friesen proved that each value appears infinitely often. - Michel Marcus, Apr 12 2019

Links

David W. Wilson, Table of n, a(n) for n = 2..10000

S. R. Finch, Class number theory [broken link]

Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Christian Friesen, On continued fractions of given period, Proc. Amer. Math. Soc. 103 (1988), 9-14.

Ron Knott, An Introduction to Continued Fractions

Formula

a(n) = A003285(A005117(n)). - Michel Marcus, Dec 29 2014

Example

a(2)=1 because 2 is the 2nd smallest squarefree integer and sqrt 2 = [ 1,2,2,2,2,... ] thus has an eventual period of 1.

Maple

sqf:= select(numtheory:-issqrfree, [$2..1000]):
map(n->nops(numtheory:-cfrac(sqrt(n), 'periodic', 'quotients')[2]), sqf); # Robert Israel, Dec 21 2014

Mathematica

Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Select[ Range[ 2, 200], SquareFreeQ] (* Harvey P. Dale, Jul 17 2011 *)

Crossrefs

Cf. A003285, A005117 (squarefree numbers), A013943.

Sequence in context: A138882 A074634 A152036 * A212829 A210215 A203647

Adjacent sequences: A035012 A035013 A035014 * A035016 A035017 A035018

Keyword

nonn,easy,nice

Author

David L. Treumann (alewifepurswest(AT)yahoo.com)

Extensions

Corrected and extended by James A. Sellers

Status

approved