A367001 Numbers that can be represented in more than one way as x^y * y^x where 1 < x <= y.

281474976710656, 150094635296999121, 1073741824000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 3587638001325335308951996778029678444520536159022591698847370237163925406936776358067367488612064034816

Offset: 1

Author

Robert Israel, Oct 31 2023

Keywords

• nonn

Comments

a(7) > 10^1500.

Includes 2^((4*k + 4)*2^((4*k + 4)*2^k + k) - 16^(2^k*(k + 1))), which is (2^x)^(2^x) * (2^x)^(2^x) = (2^(x+1))^(2^(x-k-1)) * (2^(x-k-1))^(2^(x+1)) with x = (k+1) * 2^(k+2) - 1.

Examples

			a(1) = 4^16 * 16^4 = 8^8 * 8^8 = 281474976710656
a(2) = 3^27 * 27^3 = 9^9 * 9^9 = 150094635296999121.
a(3) = 10^80 * 80^10 = 20^50 * 50^20 =
1073741824000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.
a(4) = 2^324 * 324^2 = 8^96 * 96^8 =
3587638001325335308951996778029678444520536159022591698847370237163925406936776358067367488612064034816.
a(5) = 8^144 * 144^8  = 16^96 * 96^16.
a(6) = 16^144 * 144^16 = 32^96 * 96^32.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..6

Crossrefs

Cf. A146748.

Programs

Maple

N:= 10^1000: # for terms <= N
R:= 'R': A:= NULL:
for n from 2 to ilog2(N) do
  for k from n do
    v:= n^k * k^n;
    if v > N then break fi;
    if assigned(R[v]) then A:= A,v else R[v]:= 1 fi
od od:
sort([A]);