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%I #6 Jul 13 2021 09:20:04
%S 1,11,13,19,131,5851,416833471
%N Numbers that are of both forms x^k+x+1 and x^k-(x+1) with k>=2 and x>=0.
%e 1 = 2^2-(2+1) = 0^2+(0+1)
%e 11 = 4^2-(4+1) = 2^3+(2+1)
%e 13 = 2^4-(2+1) = 3^2+(3+1)
%e 19 = 5^2-(5+1) = 2^4+(2+1)
%e 131 = 12^2-(12+1) = 5^3+(5+1)
%e 5851 = 77^2-(77+1) = 18^3+(18+1)
%e 416833471 = 20417^2-(20417+1) = 747^3+(747+1)
%p N:= 10^11: # for terms <= N
%p R:= {3}:
%p for k from 2 to ilog2(N-1) do
%p R:= R union {seq(x^k+x+1,x=2..floor(N^(1/k)))}
%p od:
%p A:= {1}:
%p for k from 2 to ilog2(N+3) do
%p for x from 2 do
%p r:= x^k-(x+1);
%p if r > N then break fi;
%p if member(r,R) then A:= A union {r} fi
%p od od:
%p sort(convert(A,list));
%Y Cf. A253913.
%K nonn,more
%O 1,2
%A _J. M. Bergot_ and _Robert Israel_, Jul 12 2021
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