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 A229851 #17 Feb 16 2025 08:33:20 %S A229851 641,114689,167772161,6597069766657,188894659314785808547841, %T A229851 850705917302346158658436518579420528641, %U A229851 2468256835981809063232453773836025757474103798450369795022913537 %N A229851 Lucky Fermat factors. %C A229851 The prime k*2^(m+2) + 1 is a lucky Fermat factor if it divides 2^(2^m) + 1 and k = 3, 5, 6, 7, or 9 is the smallest value we can choose that is not excluded by congruence constraints modulo 12, which lead to divisibility of k*2^(m+2) + 1 by 3, 5, 7, or 13 (Krizek, Luca and Somer). %C A229851 The m for which 2^(2^m) + 1 has a lucky factor are m = 5, 12, 23, 38, 73, 125, 207, 1945, 23471, 95328, 157167, 213319, 382447, 2145351, 2478782, ... From this it is trivial to write out a(1),...,a(15), but the numbers become too wide for a b-file. - _Jeppe Stig Nielsen_, Mar 13 2022 %D A229851 M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 77-79. %H A229851 Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Fermat factoring status</a> %H A229851 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a> %o A229851 (PARI) r=vector(12,m,select(k->p=k*2^(m+2)+1;p%3!=0&&p%5!=0&&p%7!=0&&p%13!=0,[3,5,6,7])[1]);for(m=0,+oo,k=r[(m+11)%12+1];p=k*2^(m+2)+1;Mod(2,p)^(2^m)+1==0&&print1(p,", ")) \\ _Jeppe Stig Nielsen_, Mar 13 2022 %Y A229851 Cf. A000215. Subsequence of A023394. %K A229851 nonn %O A229851 1,1 %A A229851 _Arkadiusz Wesolowski_, Oct 01 2013