cp's OEIS Frontend

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

%I A367001 #12 Nov 04 2023 23:18:36
%S A367001 281474976710656,150094635296999121,
%T A367001 1073741824000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,
%U A367001 3587638001325335308951996778029678444520536159022591698847370237163925406936776358067367488612064034816
%N A367001 Numbers that can be represented in more than one way as x^y * y^x where 1 < x <= y.
%C A367001 a(7) > 10^1500.
%C A367001 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.
%H A367001 Robert Israel, <a href="/A367001/b367001.txt">Table of n, a(n) for n = 1..6</a>
%e A367001 a(1) = 4^16 * 16^4 = 8^8 * 8^8 = 281474976710656
%e A367001 a(2) = 3^27 * 27^3 = 9^9 * 9^9 = 150094635296999121.
%e A367001 a(3) = 10^80 * 80^10 = 20^50 * 50^20 =
%e A367001 1073741824000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.
%e A367001 a(4) = 2^324 * 324^2 = 8^96 * 96^8 =
%e A367001 3587638001325335308951996778029678444520536159022591698847370237163925406936776358067367488612064034816.
%e A367001 a(5) = 8^144 * 144^8  = 16^96 * 96^16.
%e A367001 a(6) = 16^144 * 144^16 = 32^96 * 96^32.
%p A367001 N:= 10^1000: # for terms <= N
%p A367001 R:= 'R': A:= NULL:
%p A367001 for n from 2 to ilog2(N) do
%p A367001   for k from n do
%p A367001     v:= n^k * k^n;
%p A367001     if v > N then break fi;
%p A367001     if assigned(R[v]) then A:= A,v else R[v]:= 1 fi
%p A367001 od od:
%p A367001 sort([A]);
%Y A367001 Cf. A146748.
%K A367001 nonn
%O A367001 1,1
%A A367001 _Robert Israel_, Oct 31 2023