We (meaning the math nerds on Manifold) all know of nice numbers already: Numbers n such that n^2 and n^3 together have all possible digits with no repeats in some base.

But why limit ourselves to hypercubic numbers (i.e., squares, cubes, tesseracts, fifth powers, etc.)? What if we live in Ortholand, where people use orthoplex numbers instead? The 2-orthoplex is the square, so that one stays the same, but we will have to replace n^3 with the nth octahedral number, n(2*n^2+1)/3. To fully immerse ourselves in Ortholand, I'll denote this as n◇3 and n^2 as n◇2. So an orthonice number is a number such that n◇2 and n◇3, together, contain all the digits in some base with no repeats (and leading zeros don't count).

I've checked bases 2 through 20, and assuming I didn't make any mistakes in my math or code, there are no orthonice numbers in any of those bases. Will I find one by the end of October, or have someone else find one for me? If you find one, tell me the number and base so I can check it.