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 A076439 #17 Feb 16 2025 08:32:47
%S A076439 29,43,52,59,173,181,263,283,317,332,347,349,361,379,383,419,428,436,
%T A076439 461,467,479,484,491,509,523,529,569,571,607,613,619,641,643,653,661,
%U A076439 677,691,709,733,773,787,788,811,827,839,853,877,881,883,907,911,941
%N A076439 Numbers k which appear to have a unique representation as the difference of two perfect powers where those powers are both 2; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has x = y = 2.
%C A076439 There are two types of unique solutions. See A076438 for the general case. The k for which the unique solution can be written as k = a^2 - b^2 seems to have the following properties: (1) b = a-1 for odd k and b = a-2 for even k and (2) k = 4^r * p^s, where r is in {0,1}, p is an odd prime and s is in {1,2}. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.
%D A076439 R. K. Guy, Unsolved Problems in Number Theory, D9.
%D A076439 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
%H A076439 M. E. Bennett, <a href="http://www.math.ubc.ca/~bennett/paper19.pdf">On Some Exponential Equations Of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922.
%H A076439 T. D. Noe, <a href="http://www.sspectra.com/Pillai1a.txt">Unique solutions to Pillai's Equation requiring only squares for n <= 1000</a>.
%H A076439 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>.
%Y A076439 Cf. A001597, A076438, A076440.
%K A076439 hard,nonn
%O A076439 1,1
%A A076439 _T. D. Noe_, Oct 12 2002