This code-challenge is based on OEIS sequence A261865.
\$A261865(n)\$ is the least integer \$k\$ such that some multiple of \$\sqrt{k}\$ is in the interval \$(n,n+1)\$.
The goal of this challenge is to write a program that can find a value of \$n\$ that makes \$A261865(n)\$ as large as you can. A brute-force program can probably do okay, but there are other methods that you might use to do even better.
Example
For example, \$A261865(3) = 3\$ because
- there is no multiple of \$\sqrt{1}\$ in \$(3,4)\$ (since \$3 \sqrt{1} \leq 3\$ and \$4 \sqrt{1} \geq 4\$);
- there is no multiple of \$\sqrt{2}\$ in \$(3,4)\$ (since \$2 \sqrt{2} \leq 3\$ and \$3 \sqrt{2} \geq 4\$);
- and there is a multiple of \$\sqrt{3}\$ in \$(3,4)\$, namely \$2\sqrt{3} \approx 3.464\$.
Analysis
Large values in this sequence are rare!
- 70.7% of the values are \$2\$s,
- 16.9% of the values are \$3\$s,
- 5.5% of the values are \$5\$s,
- 2.8% of the values are \$6\$s,
- 1.5% of the values are \$7\$s,
- 0.8% of the values are \$10\$s, and
- 1.7% of the values are \$\geq 11\$.
Challenge
The goal of this code-challenge is to write a program that finds a value of \$n\$ that makes \$A261865(n)\$ as large as possible. Your program should run for no more than one minute and should output a number \$n\$. Your score is given by \$A261865(n)\$. In the case of a close call, I will run all entries on my 2017 MacBook Pro with 8GB of RAM to determine the winner.
For example, you program might output \$A261865(257240414)=227\$ for a score of 227. If two entries get the same score, whichever does it faster on my machine is the winner.
(Your program should not rely on information about pre-computed values, unless you can justify that information with a heuristic or a proof.)