cp's OEIS Frontend

A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.

%I A356879 #34 Oct 13 2022 13:58:56
%S A356879 0,2,3,8,15,18,24,32,35,48,50,63,72,80,98,99,120,128,143,162,168,195,
%T A356879 200,224,242,255,288,323,338,360,392,399,440,450,483,512,528,575,578,
%U A356879 624,648,675,722,728,783,800,840,882,899,960,968,1023,1058,1088,1152,1155,1224
%N A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.
%C A356879 Characteristics of the terms:
%C A356879   - Any x combined with any y is a solution.
%C A356879       This special case is valid only for k = 0 (exception: x = y = 0).
%C A356879   - Any x is possible and if x is odd: y = x. If x is even: y = x + 3.
%C A356879       This special case is valid only for k = 2 (see A356880).
%C A356879   - Only even x combined with y = x + 1 gives a solution.
%C A356879       Those terms are the terms of A132411.
%C A356879   - Only odd x combined with y = x gives a solution.
%C A356879       Those terms are the terms of A001105.
%C A356879   - Any x is possible and if x is odd: y = x. If x is even: y = x + 1.
%C A356879       Those terms are the terms of A132592.
%e A356879 Squares that can be produced with k = 8: 8^0 + 8^1 = 9; 8^1 + 8^1 = 16; 8^2 + 8^3 = 576; 8^3 + 8^3 = 1024; 8^4 + 8^5 = 36864; 8^5 + 8^5 = 65536; 8^6 + 8^7 = 2359296, ....
%t A356879 Select[Range[0, 1225], IntegerQ[Sqrt[# + 1]] || IntegerQ[Sqrt[#/2]] &] (* _Amiram Eldar_, Sep 18 2022 *)
%o A356879 (Python)
%o A356879 from gmpy2 import is_square
%o A356879 print([n for n in range(0,1225) if is_square(n+1) or (n % 2 == 0 and is_square(n//2))])
%Y A356879 Cf. A132411 is a subsequence (except A132411(1)), A001105 is a subsequence.
%Y A356879 Cf. A132592 is a subsequence.
%Y A356879 Cf. A356880 (k = 2), A270473 (k = 3).
%K A356879 nonn
%O A356879 0,2
%A A356879 _Karl-Heinz Hofmann_, Sep 12 2022