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A235368 The smallest number with n representations of the form A*B^A with A, B > 1.

%I A235368 #41 Feb 25 2024 10:44:16
%S A235368 8,648,344373768,
%T A235368 5797226647880997086048126220703125000000000000000000000000000000000000000000000000000000000000000000000000000
%N A235368 The smallest number with n representations of the form A*B^A with A, B > 1.
%C A235368 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.
%C A235368 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
%C A235368 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
%C A235368 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
%H A235368 Max Alekseyev, <a href="/A235368/b235368.txt">Table of n, a(n) for n = 1..6</a>
%H A235368 R. Munafo, <a href="https://mrob.com/pub/seq/cullen.html">Generalized Cullen and Woodall Numbers</a>
%H A235368 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/ABA_number/">ABA numbers</a>, Numbers Aplenty
%e A235368 a(3) = 344373768 = 8*9^8 = 3*486^3 = 2*13122^2.
%Y A235368 Cf. A171606, A171607.
%K A235368 nonn,hard
%O A235368 1,1
%A A235368 _Carlos Rivera_, Jan 08 2014
%E A235368 a(4)-a(6) from _Max Alekseyev_, Feb 21 2024