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For any relatively prime pair A and C, there are infinitely many solutions A*B^A=C*D^C, with B and D of the form B=C^W*A^X, D=C^Y*A^Z, and W,Y constrained by AW=CY+1 and X,Z constrained by CZ=AX+1. Details and examples on Munafo web page.

Call a number of the form A*B^A, A&B>1 an ABA number. Then a(n) is the smallest ABA number with n representations of this form.

a(n) exists for every n. For example, a(4) <= k = 2^105*3^70*5^126*7^120, where k (a 255-digit number) has 4 representations with A = 2, 3, 5, and 7. - Giovanni Resta, Jan 09 2014

It seems economical to find solutions where A = 2 and A = 8 both work, which is possible even though 2 and 8 are not coprime. As an example, 2^75*3^40*5^96 works for A = 2, 8, 3, 5 showing that a(4) <= 2^75*3^40*5^96 (a 109-digit number), improving the a(4) bound from previous comment. - Jeppe Stig Nielsen, Oct 29 2023

I confirm that a(4) = 2^75 * 3^40 * 5^96 with A in {2, 3, 5, 8}, a(5) = 2^315 * 3^280 * 5^336 * 7^120 with A in {2, 3, 5, 7, 8}, and a(6) = 2^1155 * 3^6160 * 5^3696 * 7^2640 * 11^2520 with A in {2, 3, 5, 7, 8, 11}. - Max Alekseyev, Feb 21 2024

Since a number can't be expressed simultaneously as 2*t^2 and 4*u^4, any number in this sequence must have a representation with A >= 5.

Up to 10^52, the only two terms not of the form 2*k^2 are 3*41841412812^3 = 4*86093442^4 = 64*3^64 and 3*54043195528445952^3 = 4*3298534883328^4 = 81*4^81.

From Charlie Neder, Jul 21 2018: (Start)

For each prime p and each value of A, the p-adic valuation of n must be congruent to the p-adic valuation of A modulo A. As a consequence, if two numbers k and m have greatest common divisor g and at least one of (k/g)^(1/g) or (m/g)^(1/g) is not an integer then no number n can have both A = k and A = m since this would lead to an unsolvable system of modular congruences.

If a number n is in this sequence with corresponding A-values {a,b,c}, then n*k^lcm(a,b,c) is also in this sequence for all k. Of the first 1198 terms, 949 of these are of the form 344373768*k^24, and 164 more are of the form 30233088000000*k^30. As values of n get larger, the proportion of primitive values rapidly decreases. (End)