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%I #33 Nov 05 2023 17:41:00
%S 344373768,30233088000000,5777633090469888,198077054103584768,
%T 97261323672455430408,5242880000000000000000,32462531054272512000000,
%U 96932598327560852471808,20526276111602783203125000,195845982777569926302400512,1631774235698698006327984128
%N Numbers expressible as A*B^A in three or more different ways, with A, B > 1.
%C 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.
%C 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.
%C From _Charlie Neder_, Jul 21 2018: (Start)
%C 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.
%C 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)
%H Charlie Neder, <a href="/A316745/b316745.txt">Table of n, a(n) for n = 1..1198</a> (terms < 10^80; terms 1..60 from Charlie Neder, terms 61..103 from Giovanni Resta)
%H G. Resta, <a href="http://www.numbersaplenty.com/set/ABA_number/">ABA numbers</a>.
%e 30233088000000 is a term because it can be expressed as 2*3888000^2 = 3*21600^3 = 5*360^5.
%t abaC[n_] := Block[{c=0, k=2}, While[n >= k 2^k, If[Mod[n, k] == 0 && IntegerQ[ (n/k)^ (1/k)], c++]; k++]; c]; lim = 10^20; a=5; Union@ Reap[ While[a 2^a < lim, b=2; While[(v = a b^a) < lim, If[abaC[v] > 2, Sow[v]]; b++]; a++]][[2, 1]]
%Y Cf. A171606, A171607, A235368.
%K nonn
%O 1,1
%A _Giovanni Resta_, Jul 12 2018
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