This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A173195 #49 Jun 12 2025 18:33:45
%S A173195 3,6,9,12,18,24,33,36,48,66,72,96,129,132,144,192,258,264,288,384,513,
%T A173195 516,528,576,768,1026,1032,1056,1152,1536,2049,2052,2064,2112,2304,
%U A173195 3072,4098,4104,4128,4224,4608,6144,8193,8196,8208,8256,8448,9216,12288
%N A173195 Values of k such that 4^x + 4^y + 4^z = k^2 with arbitrary integers x <= y <= z.
%C A173195 We prove that the solutions of 4^x + 4^y + 4^z = k^2 are (x,y,2y-x-1), for any arbitrary integer x,y. We calculate z. 4^x + 4^y + 4^z is square if positive integers m and odd integer t are such as : 1 + 4^(y-x) + 4^(z-x) = (1 + t*2^m)^2, that's why : (1 + 4^(z-y)*( 4^(y-x)) = t(1 + t*2^(m+1)) t.2^(m+1), and then m = 2y - 2x - 1. If we report this value in the precedent equation, we obtain : t-1 = (2^(z-2y+x+1) + t)(2^(z-2y+x+1) - t) * 4^(y-x-1). Because t is odd, z = 2y - x - 1. Finally, this values gives the square (2^x + 2^(2y-x-1))^2 = k^2.
%C A173195 From _Frederik P.J. Vandecasteele_, Jun 06 2025: (Start)
%C A173195 For a given n, the exponents are x = A384688(n-1), y = A138099(n), z = A000267(n-1) so that a(n) = 2^A384688(n-1) + 2^A000267(n-1).
%C A173195 Terms are all and only those k whose binary expansion is two 1 bits an odd distance apart. (End)
%D A173195 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
%D A173195 J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Ellipses, 2004.
%D A173195 H. N. Shapiro, Introduction to the Theory of Numbers, John Wiley & Sons, 1983.
%H A173195 David A. Corneth, <a href="/A173195/b173195.txt">Table of n, a(n) for n = 1..9999</a> (terms <= 10^60)
%F A173195 k = 2^x + 2^(2y-x-1), and z = 2y - x - 1.
%F A173195 Conjecture: a(n) = 3*A263132(n). - _George Beck_, May 05 2021
%e A173195 x = 0, y = 1 then z = 1, and k = 3.
%e A173195 x = 1, y = 2 then z = 2, and k = 6.
%e A173195 x = 0, y = 2 then z = 3, and k = 9.
%p A173195 for x from 0 to 1000 do :for y from x to 1000 do: n := evalf(2^x + 2^(2*y-x-1)): print (n) ; od :od :
%t A173195 Take[Union[Select[Sqrt[Flatten[Table[(2^x + 2^(2*y - x - 1))^2, {x, 0, 13}, {y, 0, 13}]]], IntegerQ]],49] (* _Jean-François Alcover_, Sep 13 2011 *)
%Y A173195 Cf. A263132.
%Y A173195 Subsequence of A018900.
%Y A173195 Cf. A384688, A138099, A000267.
%K A173195 nonn,easy
%O A173195 1,1
%A A173195 _Michel Lagneau_, Feb 12 2010