|
%I #15 Apr 20 2019 03:01:22
%S 22,523,531456,16777232,281474976710684,150094635296999160
%N Numbers representable as x^y + x + y in two or more ways, where x>1, y>1 are integers.
%C The sequence is infinite since it contains all the numbers (k^2)^(k^2-k)+k^2+k^2-k = k^(2k^2-2k)+k+2k^2-2k for k>1. - _Giovanni Resta_, May 19 2015
%C Let a, b, and k be integers such that m = ab(k^a-k^b)/(a-b) is an integer. Then, the number given by (x,y) = (k^a,m/a) is the same as that given by (k^b,m/b). The given terms correspond to (a,b,k) = (2,1,2), (3,1,2), (2,1,3), (3,2,2), (4,2,2)/(2,1,4), and (3,1,3). - _Charlie Neder_, Apr 19 2019
%e a(1) = 22 = 2^4 + 2 + 4 = 4^2 + 4 + 2.
%e a(2) = 523 = 8^3 + 8 + 3 = 2^9 + 2 + 9.
%e a(3) = 531456 = 3^12 + 3 + 12 = 9^6 + 9 + 6.
%e a(4) = 16777232 = 4^12 + 4 + 12 = 8^8 + 8 + 8.
%e a(5) = 281474976710684 = 4^24 + 4 + 24 = 16^12 + 16 + 12.
%e a(6) = 150094635296999160 = 3^36 + 3 + 36 = 27^12 + 27 + 12.
%Y Cf. A253775, A253776.
%K nonn,hard,more
%O 1,1
%A _Alex Ratushnyak_, Jan 12 2015
%E a(6) from _Lars Blomberg_, May 19 2015
|
