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 A323605 #45 Nov 02 2023 14:19:19 %S A323605 2,3,7,43,13,3263443,547,29881,5295435634831,181,2287,73 %N A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence). %C A323605 a(n) is also the smallest prime divisor of A007018(n+1) that is not a divisor of A007018(n). %C A323605 The prime numbers a(n) are all distinct, which proves the infinitude of the prime numbers (Saidak's proof). %C A323605 a(12) <= 2589377038614498251653. - _Daniel Suteu_, Jan 20 2019 %C A323605 a(12)..a(50) = [?, 52387, 13999, 17881, 128551, 635263, ?, ?, 352867, 387347773, ?, 74587, ?, ?, 27061, 164299, 20929, 1171, ?, 1679143, ?, ?, 120823, 2408563, 38218903, 333457, 30241, 4219, 1085443, 7603, 1861, ?, 23773, 51769, 1285540933, 429547, ?, 8323570543, ?], where ? denote unknown values > 10^10. - _Max Alekseyev_, Oct 11 2023 %H A323605 Filip Saidak, <a href="http://dx.doi.org/10.2307/27642094">A new proof of Euclid's theorem</a>, Amer. Math. Monthly, 113:10 (2006) 937-938. %H A323605 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sylvester%27s_sequence#Divisibility_and_factorizations">Sylvester's sequence: Divisibility and factorizations</a>. %F A323605 a(n) = A007996(m), where m is the smallest index such that A180871(m) = n. - _Max Alekseyev_, Oct 11 2023 %p A323605 with(numtheory): %p A323605 u:=1: P:=NULL: to 9 do P:=P,sort([op(divisors(u+1))])[2]: u:=u*(u+1) od: %p A323605 P; %o A323605 (PARI) f(n)=if(n<1, n>=0, f(n-1)+f(n-1)^2); \\ A007018 %o A323605 a(n)=divisors(f(n)+1)[2]; \\ _Michel Marcus_, Jan 20 2019 %Y A323605 Cf. A007018, A000058, A007996, A091335, A180871. %K A323605 nonn,more %O A323605 0,1 %A A323605 _Robert FERREOL_, Jan 19 2019 %E A323605 a(10)-a(11) from _Daniel Suteu_, Jan 20 2019