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

8, 648, 344373768, 5797226647880997086048126220703125000000000000000000000000000000000000000000000000000000000000000000000000000

Offset: 1

Author

Carlos Rivera, Jan 08 2014

Keywords

• nonn
• hard

Comments

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

Examples

			a(3) = 344373768 = 8*9^8 = 3*486^3 = 2*13122^2.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..6
• R. Munafo, Generalized Cullen and Woodall Numbers
• Giovanni Resta, ABA numbers, Numbers Aplenty

Crossrefs

Cf. A171606, A171607.

Extensions

a(4)-a(6) from Max Alekseyev, Feb 21 2024