A215485 -id:A215485 - cp's OEIS frontend

A215508 Smallest m such that the period of the continued fraction of sqrt(m) is A215485(n); records of A013646.

1, 2, 3, 41, 58, 106, 193, 337, 586, 949, 1061, 1117, 1153, 1249, 1669, 2381, 3733, 5857, 6577, 6781, 8389, 11173, 14293, 15817, 17137, 17209, 23017, 37921, 38377, 46261, 47293, 56929, 82561, 90121, 113173, 122401, 148957, 151057, 161149, 163729, 193873, 206209, 225769, 322513, 497473, 576529, 676129, 686893, 706621, 862921, 946489, 992281, 1032649, 1198081, 1597033, 1655677, 1779409, 1930021, 2299489, 2367481, 2584081, 3209281, 3528409, 3933073, 4068241, 4160521, 4283689, 4726009, 4833901

Offset: 0

Patrick McKinley, Aug 13 2012

nonn

The continued fractions of these numbers have the "hard to get" lengths listed in sequence A215485. They fill the last gaps in the table when computing A013646.

			The lengths of the continued fractions of sqrt(1), sqrt(2), sqrt(3) and sqrt(41) are 0, 1, 2 and 3 respectively. The rest of the sequence follows A215485 similarly.

Patrick McKinley, Table of n, a(n) for n = 0..253

Cf. A013646, A215485.

a(n) = A013646(A215485(n)). - Pontus von Brömssen, Nov 24 2024

A013646 Least m such that the continued fraction for sqrt(m) has period n.

Original entry on oeis.org

1, 2, 3, 41, 7, 13, 19, 58, 31, 106, 43, 61, 46, 193, 134, 109, 94, 157, 139, 337, 151, 181, 166, 586, 271, 457, 211, 949, 334, 821, 379, 601, 463, 613, 331, 1061, 478, 421, 619, 541, 526, 1117, 571, 1153, 604, 1249, 694, 1069, 631, 1021, 1051, 1201, 751, 1669, 886

Offset: 0

N. J. A. Sloane, Clark Kimberling, Walter Gilbert

In a search of fractions up to sqrt(1650241399), the smallest length not yet seen is 97921. The next unseen lengths are 101679, 102181 and 102407. After 145 more missing odd lengths, the first even length not seen is 107292. This would suggest that A215485 may be exclusively odd after an early 2, but beware the law of small numbers! - Patrick McKinley, Aug 24 2012
a(97921) = 1664155249, a(101679) = 1654486681, a(102181) = 1682919001, a(102407) = 1680133849, a(107292) = 1651931884, thus 107292 is not in A215485. - Chai Wah Wu, Jun 08 2017
a(999213) = 133511789629, a(1000000) = 98814608764. - Michael Hortmann, Mar 20 2023

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

Patrick McKinley, Table of n, a(n) for n = 0..97920 (first 1000 terms from T. D. Noe)
Michael Hortmann, Table of n, a(n) for n = 0..1000000

Cf. A003285, A215485.

a[n_] := Catch[For[m = 1, True, m++, If[Length[ Last[ ContinuedFraction[ Sqrt[m] ]]] == n, Print[m]; Throw[m] ]]]; Table[a[n], {n, 0, 54}](* Jean-François Alcover, May 15 2012 *)
Flatten[Table[Position[Table[{s=Sqrt[n]};If[IntegerQ[s],0,Length[ ContinuedFraction[s] [[2]]]], {n,2000}],i,{1},1],{i,0,60}]] (* Harvey P. Dale, Sep 15 2013 *)

A003285(a(n)) = n. - Pontus von Brömssen, Nov 24 2024

cp's OEIS Frontend

A215508 Smallest m such that the period of the continued fraction of sqrt(m) is A215485(n); records of A013646.

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