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 A386311 #26 Jul 23 2025 19:38:21
%S A386311 2,3,13,29,71,73,3673,3677,1970327,8879341,30578677
%N A386311 a(1) = 2, a(n+1) is the largest prime p such that b(n+1) = b(n)*(p + a(n))/(p - a(n)) is a positive integer, where b(1) = 1.
%C A386311 This sequence is finite and full.
%C A386311 Note that a(n) < a(n+1) <= 2*b(n) + a(n).
%C A386311 b(1) = 1, b(n+1) is the smallest k such that a(n+1) = a(n)*(k + b(n))/(k - b(n)) is a prime, where a(1) = 2.
%C A386311 b(n) = 1, 5, 8, 21, 50, 3600, 3746, 6883275, 6909014, 10849668, and 19729009.
%C A386311 Conjecture: a'(n) = prime(n) for "the smallest prime p" and b'(n) = A352743(n-1) for "the largest k".
%C A386311 If a(n+1) <= 2*b(n) + a(n), then a(11) = 30578677 is the last term. - _M. F. Hasler_, Jul 19 2025
%F A386311 Product_{k=1..n} (a(k+1) + a(k))/(a(k+1) - a(k)) = b(n+1).
%F A386311 a(n+1)/a(n) = (b(n+1) + b(n))/(b(n+1) - b(n)).
%F A386311 b(n+1)/b(n) = (a(n+1) + a(n))/(a(n+1) - a(n)).
%o A386311 (PARI) {a=List(2); b=List(1); for(n=1,oo, print1(a[n]", "); my(an=a[n], bn=b[n], p=precprime(2*bn+an)); iferr(while(bn*(p+an)%(p-an), p=precprime(p-1)), E, break); listput(a, p); listput(b, bn*(p+an)\(p-an))); print("that's all."); a=Vec(a)} \\ _M. F. Hasler_, Jul 19 2025
%Y A386311 Cf. A000040, A352743 (see author's conjecture).
%K A386311 nonn,fini,full
%O A386311 1,1
%A A386311 _Thomas Ordowski_, Jul 18 2025
%E A386311 a(7)-a(10) from _M. F. Hasler_, Jul 18 2025
%E A386311 a(11) = 2*b(10)+a(10) from _Thomas Ordowski_, Jul 19 2025