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

Author

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

Keywords

• nonn
• easy
• nice

Comments

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

Examples

			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.
		

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

Crossrefs

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

Programs

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 *)

Formula

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

Extensions

Corrected and extended by James Sellers