A253777 Numbers representable as x^y + x + y in two or more ways, where x>1, y>1 are integers.

22, 523, 531456, 16777232, 281474976710684, 150094635296999160

Offset: 1

Author

Alex Ratushnyak, Jan 12 2015

Keywords

• nonn
• hard
• more

Comments

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

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

Examples

			a(1) = 22 = 2^4 + 2 + 4 = 4^2 + 4 + 2.
a(2) = 523 = 8^3 + 8 + 3 = 2^9 + 2 + 9.
a(3) = 531456 = 3^12 + 3 + 12 = 9^6 + 9 + 6.
a(4) = 16777232 = 4^12 + 4 + 12 = 8^8 + 8 + 8.
a(5) = 281474976710684 = 4^24 + 4 + 24 = 16^12 + 16 + 12.
a(6) = 150094635296999160 = 3^36 + 3 + 36 = 27^12 + 27 + 12.
		

Crossrefs

Cf. A253775, A253776.

Extensions

a(6) from Lars Blomberg, May 19 2015