A Prolific Pair: 364585 and 820359

What do the numbers 364585 and 820359 have in common? Well, they're both odd for starters. However, although it's not obvious, they are both prolific in the sense that when squared, they produce numbers that have seven sequential, identical, non-zero digits (permalink):$$ \begin{align} 364585^2 &= 132922222225\\820359^2 &= 672988888881 \end{align}$$They are the only numbers with this property in the range up to one million. The first number mentioned (364585) makes an appearance in OEIS  A167712 for the case of \(n=7\):

A167712

\(a(n) \) = the smallest positive number, not ending in 0, whose square has a substring of exactly \(n\) identical digits.

The sequence begins: 1, 12, 38, 1291, 10541, 57735, 364585, 1197219, 50820359, 169640142, 298142397, 4472135955, 1490711985, 2185812841434

This same number makes another appearance in OEIS A048347 for the case of \(n=8\) where it should be noted that the identical digits are not necessarily sequential

A048347

\(a(n)^2\) is the smallest square containing exactly \(n\) 2's.

The sequence begins: 5, 15, 149, 1415, 4585, 14585, 105935, 364585, 3496101, 4714045, 34964585, 149305935, 1490725415, 4714469665, 1490711985, 149071333335, 1105537083332, 1489973900149, 15106363633335, 47140462469223

If we look at the cubes of numbers, then not surprisingly more numbers satisfy, specifically (permalink):$$ \begin{align} 339247^3 &= 39043437522222223 \\475741^3 &= 107674222222294021 \\ 720993^3 &= 374794444444986657 \\ 822389^3 &= 556201144444449869 \\ 699637^3 &= 342466666667067853 \\ 360598^3 &= 46888888826167192 \\ 948083^3 &= 852195188888887787 \\985426^3 &= 956912108888888776 \\ 764149^3 &= 446204706999999949 \end{align} $$So nothing profound in this post, just some peculiarities of the base 10 number system that produce runs of seven identical non-zero digits for certain squares and cubes of numbers up to one million.