
COMMENTS

Prime values for n = 2,4,6,8, with no others up to n = 3400. Beiler mentions this pattern in the reference.
From Peter Bala, Sep 27 2015: (Start)
Calculation suggests the continued fraction expansion of sqrt(a(n)), for n >= 1, begins [10^n  1, 1, 1, 1/3*(2*10^n  5), 1, 5, 1/9*(2*10^n  11), 1, 17, (2*10^n  20  9*(1  MOD(n, 3)))/27, ...]. Note the large partial quotients early in the expansion.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions. Empirically, we also see exceptionally large partial quotients in the continued fraction expansions of the mth root of the numbers a(m*n) for m = 2, 3, 4,... as n increases. Some examples are given below. Cf. A000533, A002283, A066138. (End)


EXAMPLE

Simple continued fraction expansions showing large partial quotients:
sqrt(a(10)) = [9999999999; 1, 1, 6666666665, 1, 5, 2222222221, 1, 17, 740740740, 1, 1, 1, 5, 2, 1, 246913579, 1, 1, 4, 1, 1, 3, 1, 1, ...].
a(18)^(1/3) = [999999999999; 1, 2999999, 499999999999, 1, 1439999, 2582644628099, 5, 1, 3, 4, 1, 58, 1, 1, 1, 8, ...].
a(30)^(1/5) = [999999999999; 1, 4999999999999999999, 333333333333, 3, 217391304347826086, 1, 1, 1, 1, 1, 8, 2398081534, 1, 1, 1, 9, 1, 98, 1, 125052522059263, 1, 9, 7, 1, ...].  Peter Bala, Sep 27 2015
