The divisor matrix $D=(d_{r,s})_{r,s\in\mathbf{N}}$ is defined by $d_{r,s}=1$ if $r$ divides $s$ and $0$ otherwise. Raymond Redheffer considered a finite truncation of the divisor matrix. For each natural number $n$ he considered the matrix, denoted $R_{n}$, obtained from the upper left $n\times n$ submatrix of $D$ by setting each entry in the first column equal to 1. I say $R_{n}$ is even if its number of nonzero entries is even, otherwise I say it is odd. Consider the following table: \begin{array}{ l | l | l } n & \text{No. of nonzero entries of $R_{n}$} & \text{parity of $R_{n}$} \\ \hline 1 & 1 & \text{odd} \\ 2 & 4 & \text{even} \\ 3 & 7 & \text{odd} \\ 4 & 11 & \text{odd} \\ 5 & 14 & \text{even} \\ 6 & 19 & \text{odd} \\ 7 & 22 & \text{even} \\ 8 & 27 & \text{odd} \\ 9 & 31 & \text{odd} \\ 10 & 36 & \text{even} \\ 11 & 39 & \text{odd} \\ 12 & 46 & \text{even} \\ 13 & 49 & \text{odd} \\ 14 & 54 & \text{even} \\ 15 & 59 & \text{odd} \\ \end{array} The values $n$ for which the matrices $R_{n}$, are even in parity are the numbers $2,5,7,10,12,14,\ldots$ Similarly the values $n$ for which the matrices $R_{n}$ are odd in parity are the numbers $1,3,4,6,8,9,11,13,15,\ldots$
Writing in 1959 to the journal American Mathematical Monthly Ian Connell challenged readers to find a closed form to an unusual sequence. The sequence, which now bears his name, is made as follows: begin by writing the first odd number, $1$, then write the next two even numbers after $1$: $2, 4$ and then write the next three odd numbers after $4$: $5, 7, 9$ and then write the next four even numbers after $9$: $10, 12, 14, 16$, and so on. The first few terms are $1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17,\ldots$ A solution to Connell’s challenge appeared in the same journal in 1960, where it was shown the $n$-th term of the Connell sequence, $a_{n}$, has the closed form $2n − \lfloor1/2(1 + \sqrt{8n − 7})\rfloor$. The sequences is found in the Online Encyclopedia of Integer Sequences (OEIS) by looking up the entry A001614. A modification to the Connell Sequence can be made by replacing the first term of the Connell sequence with the even number $0$ and then proceed by writing down runs of $n$ consecutive numbers of a given parity. In particular begin by writing the first even number $0$, then write the next two odd numbers after $0$: $1, 3$ then write the next three even numbers after $3$: $4, 6 ,8$ and write the next four odd numbers after $8$: $9,11,13, 15$, and so on. The first few terms are $0,1,3,4,6,8,9,11,13,15,\ldots$ The modified Connell sequence is found in the OEIS by looking up the entry A133280. There one finds a closed formula for the $n$-th term, $a_{n}$, namely $1+2n−\lfloor 1/2+\sqrt{2n + 2}\rfloor$.
I say a number is a Connell number if it belongs to the Connell sequence. I say a number is a modified Connell number if it belongs to the modified Connell sequence. The Connell sequence and their modifications span the natural numbers and share the perfect squares between them. It is straightforward to show that the average order of the Connell numbers and their modification is $2,$ that is $\lim_{n\to\infty}a_{n}/n=2.$
Here is the Question:
Are the following items true?
- $R_{n}$ is even if and only if $n$ is a Connell number that is not a perfect square.
- $R_{n}$ is odd if and only if $n$ is a modified Connell number.
For example $n=19$ is a Connell number that is not a perfect square, and so we should expect $R_{19}$ to have an even number on nonzero entries. Indeed, the number of nonzero entries in $R_{19}$ is 78. Similarly, if $n=36$, a modified Connell number then we should expect $R_{36}$ to have an odd number of nonzero entries. Sure enough, $R_{36}$ has 175 nonzero entries.
