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 A206418 #32 Apr 03 2023 10:36:13
%S A206418 2,22,127,55,1527,18453,5517
%N A206418 a(n) is the least integer k > 1 such that 1 + k^(5^n) + k^(2*5^n) + k^(3*5^n) + k^(4*5^n) is prime.
%C A206418 Phi(5^(n+1), k) = 1 + k^(5^n) + k^(2*5^n) + k^(3*5^n) + k^(4*5^n).
%C A206418 The primes correspond to k(1) through k(4) have a p-1 factorable up to 34% or higher, thus are proved prime by OpenPFGW.
%C A206418 The fifth one, Phi(5^6,18453) = 1 + 18453^3125 + 18453^6250 + 18453^9375 + 18453^12500, is a 55326-digit Fermat and Lucas PRP with 78.86% proof. A CHG proofing is running but it will take month to complete.
%C A206418 The sixth one, Phi(5^7,5517), has 233857 digits and can only be factored to about 26%. It is too big for CHG to provide a proof.
%H A206418 David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/chgcertd.gp">Coppersmith--Howgrave-Graham certificate tester</a> (2006)
%H A206418 Chris Caldwell, <a href="https://t5k.org/bios/page.php?id=797">John Renze's Coppersmith-Howgrave-Graham PARI script</a>
%F A206418 a(n) = A085398(5^(n+1)). - _Jinyuan Wang_, Dec 21 2022
%e A206418 Phi(5^2, 22) = 705429635566498619547944801 is prime, while Phi(25, k) with k = 2 to 21 are composites, so a(1) = 22.
%t A206418 Table[i = 1; m = 5^u; While[i++; cp = 1 + i^m + i^(2*m) + i^(3*m) + i^(4^m); ! PrimeQ[cp]]; i, {u, 1, 4}]
%o A206418 (PARI) See Broadhurst link.
%o A206418 (PARI) a(n)=my(k=2);while(!ispseudoprime(polcyclo(5,k^n)),k++);k \\ _Charles R Greathouse IV_, Feb 09 2012
%Y A206418 Cf. A085398, A153438.
%K A206418 nonn,hard,more
%O A206418 0,1
%A A206418 _Lei Zhou_, Feb 09 2012
%E A206418 a(0) inserted by _Jinyuan Wang_, Dec 21 2022