Ron Graham's sequence is a neat bijection from the positive integers to the non-primes, defined as following:
$a(n)$ = smallest $m$ for which there is a sequence $n = b_1 < b_2 < \dotsc < b_t = m$ such that $b_1 \cdot b_2 \cdot \dotsc \cdot b_t$ is a perfect square.
Here are some examples:
n | a(n) | example sequence | minimum t | product
--+------+------------------+-----------+---------
1 | 1 | [1] | 1 | 1^2
2 | 6 | [2, 3, 6] | 3 | 6^2
3 | 8 | [3, 6, 8] | 3 | 12^2
4 | 4 | [4] | 1 | 2^2
5 | 10 | [5, 8, 10] | 3 | 20^2
6 | 12 | [6, 8, 12] | 3 | 24^2
7 | 14 | [7, 8, 14] | 3 | 28^2
8 | 15 | [8, 10, 12, 15] | 4 | 120^2
I'm interesting in finding proof for any of the following (equivalent) conjectures that have been made about this sequence:
- $n \cdot a(n)$ is nonsquare for all nonsquare $n$.
- The minimum value of $t$ is never equal to $2$. (See A066440.)
- Let $b(n)$ be the least integer $k > n$ such that $kn$ is square. Then $b(n) - a(n) > 0$. (See A255167.)
Scatterplot of $b(n) - a(n)$.
Notice that there is not an obvious lower bound.
Scatterplot of the minimum values of $t$.
Notice that $A066440(n) \neq 2$ for the first 10,000 terms.
