A062769 -id:A062769 - cp's OEIS frontend

A240072 Least number k with continued fraction of sqrt(k) having periodic part of length 2*n.

3, 7, 19, 31, 43, 46, 134, 94, 139, 151, 166, 271, 211, 334, 379, 463, 331, 478, 619, 526, 571, 604, 694, 631, 1051, 751, 886, 1039, 1141, 919, 1291, 1324, 1699, 1879, 1366, 2476, 2038, 1516, 1894, 1759, 2164, 1831, 2179, 1726, 2851, 2461, 2011, 2311, 4603

Offset: 1

T. D. Noe, Apr 04 2014

nonn

It appears that, in general, these numbers are less than the corresponding numbers for the odd lengths, A062769.

			The continued fractions of sqrt(3), sqrt(7), and sqrt(19) are {1; 1, 2}, {2; 1, 1, 1, 4}, and {4; 2, 1, 3, 1, 2, 8}.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A013646 (even and odd), A062769 (similar, but odd length).

nn = 50; t = Table[0, {nn}]; n = 1; found = 0; While[found < nn, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[EvenQ[len] && len/2 <= nn && t[[len/2]] == 0, t[[len/2]] = n; found++]]]; t

A288184 Least odd number k such that the continued fraction for sqrt(k) has period n.

Original entry on oeis.org

5, 3, 41, 7, 13, 19, 73, 31, 113, 43, 61, 103, 193, 179, 109, 133, 157, 139, 337, 151, 181, 253, 853, 271, 457, 211, 949, 487, 821, 379, 601, 463, 613, 331, 1061, 1177, 421, 619, 541, 589, 1117, 571, 1153, 823, 1249, 739, 1069, 631, 1021, 1051, 1201, 751

Offset: 1

Ilya Gutkovskiy, Jun 06 2017

nonn

			a(2) = 3, sqrt(3) = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/...)))), period 2: [1, 2].

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Periodic Continued Fraction

Cf. A000035, A003285, A010337-A010339, A013642-A013644, A013646, A013943, A020347-A020439, A059800, A062769, A097853, A288185.

from sympy import continued_fraction_periodic
def A288184(n):
    d = 1
    while True:
        s = continued_fraction_periodic(0,1,d)[-1]
        if isinstance(s, list) and len(s) == n:
            return d
        d += 2 # Chai Wah Wu, Jun 07 2017

A003285(a(n)) = n, A000035(a(n)) = 1.

cp's OEIS Frontend

A240072 Least number k with continued fraction of sqrt(k) having periodic part of length 2*n.

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