A316745 Numbers expressible as A*B^A in three or more different ways, with A, B > 1.

344373768, 30233088000000, 5777633090469888, 198077054103584768, 97261323672455430408, 5242880000000000000000, 32462531054272512000000, 96932598327560852471808, 20526276111602783203125000, 195845982777569926302400512, 1631774235698698006327984128

Offset: 1

Author

Giovanni Resta, Jul 12 2018

Keywords

• nonn

Comments

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)

Examples

			30233088000000 is a term because it can be expressed as 2*3888000^2 = 3*21600^3 = 5*360^5.
		

Links

• Charlie Neder, Table of n, a(n) for n = 1..1198 (terms < 10^80; terms 1..60 from Charlie Neder, terms 61..103 from Giovanni Resta)
• G. Resta, ABA numbers.

Crossrefs

Cf. A171606, A171607, A235368.

Programs

Mathematica

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]]