A346287 Numbers that are of both forms x^k+x+1 and x^k-(x+1) with k>=2 and x>=0.

1, 11, 13, 19, 131, 5851, 416833471

Offset: 1

Author

J. M. Bergot and Robert Israel, Jul 12 2021

Keywords

• nonn
• more

Examples

			1 = 2^2-(2+1) = 0^2+(0+1)
11 = 4^2-(4+1) = 2^3+(2+1)
13 = 2^4-(2+1) = 3^2+(3+1)
19 = 5^2-(5+1) = 2^4+(2+1)
131 = 12^2-(12+1) = 5^3+(5+1)
5851 = 77^2-(77+1) = 18^3+(18+1)
416833471 = 20417^2-(20417+1) = 747^3+(747+1)
		

Crossrefs

Cf. A253913.

Programs

Maple

N:= 10^11: # for terms <= N
R:= {3}:
for k from 2 to ilog2(N-1) do
  R:= R union {seq(x^k+x+1,x=2..floor(N^(1/k)))}
od:
A:= {1}:
for k from 2 to ilog2(N+3) do
  for x from 2 do
    r:= x^k-(x+1);
    if r > N then break fi;
    if member(r,R) then A:= A union {r} fi
od od:
sort(convert(A,list));