Certain things guessed about by me on the subject of a^b and its counts of digits are obviously logically, mathematically false (with a little careful thought). There should eventually be a last power of any kind that has all 10 digits counted a prime number of times, essentially as intersections of 0-density sets of integers (without intimate linkage, correlating cause). I would ask what the largest exponent on such a number could be, as marginally to thoroughly tractable (for making a sound conjecture). The largest square, though it is sure to exist, is not findable in my opinion, and is most likely the largest power to have the other criterion.

For numbers of the form p^q (both primes), this is a sequence whose largest currently known member is the largest number of the kind I know so far:

{10005835517^2, 4697081^3, 52181^5, 1019^7, 5087^11, 347^13, 2029^17, 7351^19, 23^23, 1554841^29, 526297^31, 8843729^37, 10336567^41, 17650603^43, ...}

For search of reasonable-size exponent, and increasing prime base without limiting the exponent, a record size is set by 2029^17, with 2029 as the 13th prime to give a value, and this size record is broken first again by 3343^18, with 3343 as the 21st such prime (2029 is 4th--after 23, 347 and 1019--to have prime exponent first for this (or at all, most likely)).

FWIW, right now I don't remember the specific question that 69636^4 answered. It is written down. Also, cute of the type is 18181^8, but the number of numbers satisfying the criterion is pretty large, so this one is just something that happened to catch my eye.

The smallest length for powers of numbers not divisible by 3 is necessarily 21, and perhaps an interesting question is how many powers of primes of this type there are of exactly this length. I may try to say later.

JGM

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