I recently discovered an interesting connection between the following two On-Line Encyclopedia of Integer Sequences (OEIS) sequences: A001001 and A209635. More specifically, there seems to be an interesting connection between the reduction of A001001 modulo $2$ and the OEIS sequence A209635.
Let A001001$(n)$ denote the number of sublattices of index $n$ in generic 3-dimensional lattice.
Let A000265$(n)$ denote the largest odd divisor of $n$.
Reduce the sequence A001001 modulo 2, yielding:
$$(1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, \ldots)$$
Now compute the integer sequence $(|\mu($A000265$(n))| : n \in \mathbb{N} )$:
$$(1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, \ldots)$$
We have that A001001$(n)(\text{mod $2$}) = |\mu($A000265$(n))| $ for $n = 1, 2, \ldots 80$.
We have that A001001$(n)(\text{mod $2$})$ is not equal to $|\mu($A000265$(n))| $ if and only if $n$ is an entry in the following integer sequence, which we denote by $(a_{n})_{n \in \mathbb{N}}$ (assuming this sequence is infinite):
$$(81, 162, 243, 324, 405, 486, 567, 625, 648, 810, 891, 972, 1053, 1134, 1215, 1250, 1296, \ldots)$$
This integer sequence is not currently in the OEIS. Now consider the sequence $(\frac{a_{n}}{n})_{n \in \mathbb{N}}$:
$$(81,81,81,81,81,81,81,\frac{625}{8},72,81,81,81,81,81,81,\frac{625}{8},\frac{1296}{17},\frac{153}{2},81,81,81,81, \ldots)$$
This intuitively seems to be a very strange sequence, since the integer $81$ at first seems to appear frequently but not in a 'predictable' way.
I have several questions regarding the connection between A001001 and A209635 described above.
Question 1: Is there any intuitive number-theoretic explanation as to 'why' the parity of A001001 is so similar to the integer sequence $(|\mu($A000265$(n))| : n \in \mathbb{N} )$?
Question 2: What is $(a_{n})_{n \in \mathbb{N}}$?
Question 3: What is $\lim_{n \to \infty}\frac{a_{n}}{n}$? Some plots illustrating the apparent convergence of the sequence $\left( \frac{a_{n}}{n} : n \in \mathbb{N}\right)$ are given below.
