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 A321518 #22 Dec 23 2024 14:53:45 %S A321518 3,2,0,24 %N A321518 Smallest k > 1 such that n^k + k^n is prime, i.e., a Leyland prime, or 0 if no such k exists. %C A321518 a(4) = 0. Proof: For k == 1 (mod 4), 4^k + k^4 = 4*x^4 + k^4 = (2*x^2 - 2*k*x + k^2)(2*x^2 + 2*k*x + k^2), where x = 4^((k-1)/4). For k == 3 (mod 4), 4^k + k^4 = 64*x^4 + k^4 = (8*x^2 - 4*k*x + k^2)(8*x^2 + 4*k*x + k^2), where x = 4^((k-3)/4) (cf. Israel, 2015). %C A321518 Conjecture: a(6) = 0. %C A321518 From _Jon E. Schoenfield_, Nov 13 2018: (Start) %C A321518 Let t = 6^k + k^6. %C A321518 If k is even, then 2|t. %C A321518 If k is odd but not divisible by 7, then 7|t. %C A321518 If k is divisible by 3, then 3|t. %C A321518 If k == 7 or 63 (mod 70), then 5|t. %C A321518 Thus, a(6) == 35, 49, 91, 119, 161, or 175 (mod 210) if a(6) > 0. (End) %H A321518 Robert Israel, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-December/015820.html">Re: Smallest k > 1 such that n^k+k^n is prime, or 0 if no such k exists</a>, Seqfan (Dec 11 2015). %H A321518 Wikipedia, <a href="https://en.wikipedia.org/wiki/Leyland_number">Leyland number</a> %e A321518 For n = 5: 5^24 + 24^5 = 59604644783353249 is prime, and 24 is the smallest k > 1 such that 5^k + k^5 is prime, so a(5) = 24. %Y A321518 Cf. A076980, A094133. %K A321518 nonn,hard,more %O A321518 2,1 %A A321518 _Felix Fröhlich_, Nov 12 2018