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 A068563 #32 Apr 19 2025 06:15:32 %S A068563 1,2,4,6,8,12,16,18,20,24,32,36,40,42,48,54,60,64,72,80,84,96,100,108, %T A068563 120,126,128,136,144,156,160,162,168,180,192,200,216,220,240,252,256, %U A068563 272,288,294,300,312,320,324,336,342,360,378,384,400,408,420,432,440 %N A068563 Numbers k such that 2^k == 4^k (mod k). %C A068563 If k is in the sequence then 2k is also in the sequence, but the converse is not true. %C A068563 Contains A124240 as a subsequence. Their difference is given by A124241. - _T. D. Noe_, May 30 2003 %C A068563 Also, integers k such that A007733(k) divides k. Also, integers k such that for every odd prime divisor p of k, A007733(p) = A002326((p-1)/2) divides k. Also, integers k such that A000265(k) divides 2^k-1. - _Max Alekseyev_, Aug 25 2013 %H A068563 Amiram Eldar, <a href="/A068563/b068563.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe) %H A068563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelFunction.html">Carmichael Function</a>. %t A068563 Select[Range[500], PowerMod[2,#,# ] == PowerMod[4,#,# ] & ] %o A068563 (PARI) isok(k) = Mod(2, k)^k == Mod(4, k)^k; \\ _Amiram Eldar_, Apr 19 2025 %Y A068563 Cf. A000265, A002322, A002326, A007733, A124240, A124241. %K A068563 easy,nonn %O A068563 1,2 %A A068563 _Benoit Cloitre_, Mar 25 2002 %E A068563 Comment and Mathematica program corrected by _T. D. Noe_, Oct 17 2008