OFFSET
1,2
COMMENTS
Conjecture: A003285(m) = even or A004613, if m is divisible by A003285(m). [This sentence appears to be saying that all odd terms of this sequence are in A004613.]
Because A003285(m) < 3.76*sqrt(m)*log(m) (see Stanton et al.), it is enough to check m such that m <= (3.76*n*log(m))^2. For n <= 36 it even suffices to check m <= 5916*n. - Nathaniel Johnston, May 10 2011
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..406
R. G. Stanton, C. Sudler, and H. C. Williams, An upper bound for the period of the simple continued fraction for sqrt(D), Pacific Journal of Mathematics, Vol. 67, No. 2 (1976), pp. 525-536.
MAPLE
with(numtheory): A065129 := proc(n) local m: if(n=1)then return 0:fi: for m from n by n to 5916*n do if(frac(sqrt(m))<>0)then if(n*nops(cfrac(sqrt(m), 'periodic', 'quotients')[2])=m)then return m: fi: fi: od: return 0: end: seq(A065129(n), n=1..10); # Nathaniel Johnston, May 10 2011
MATHEMATICA
Do[k = 2; While[ k / Length[ Last[ ContinuedFraction[ Sqrt[k]]]] != n, k++ ]; Print[k], {n, 2, 10} ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Naohiro Nomoto, Nov 14 2001
EXTENSIONS
a(11)-a(37) from Nathaniel Johnston, May 10 2011
Terms a(38) and beyond from Chai Wah Wu, Jan 27 2021
STATUS
approved