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 A321616 #39 Aug 21 2024 11:19:09
%S A321616 5,61,113,1741
%N A321616 Primes p = k^2 + (k-1)^2 such that k^p - (k-1)^p is prime.
%C A321616 Conjecture: generally, these are primes p = a^2 + b^2 with a > b > 0 such that (a^p - b^p)/(a-b) is prime, so must be a-b = 1. It seems that there are no primes (a^q + b^q)/(a+b) for primes q = a^2 + b^2 > 5. Especially, there are probably no primes q = m^2 + 1 > 5 such that (m^q - 1)/(m-1) is prime or (m^q + 1)/(m+1) is prime. How to prove it?
%C A321616 No more terms up to the prime 19801 = 100^2 + 99^2. - _Amiram Eldar_, Nov 15 2018
%C A321616 a(5) > 109045. - _J.W.L. (Jan) Eerland_, Dec 11 2022
%C A321616 a(5) > 209305. - _Michael S. Branicky_, Aug 21 2024
%e A321616 The prime 5 = 2^2 + 1^2 and 2^5 - 1^5 = 31 is prime.
%e A321616 We have 61 = 6^2 + 5^2, 113 = 8^2 + 7^2, 1741 = 30^2 + 29^2.
%t A321616 f[k_]:=k^2 + (k-1)^2 ; seqQ[k_]:=Module[{p=f[k]}, PrimeQ[p] && PrimeQ[k^p - (k-1)^p ]]; f[Select[Range[30], seqQ]] (* _Amiram Eldar_, Nov 15 2018 *)
%t A321616 pQ[k_]:=Module[{c=k^2+(k-1)^2},If[AllTrue[{c,k^c-(k-1)^c},PrimeQ],c,Nothing]]; Array[pQ,30] (* _Harvey P. Dale_, Aug 27 2023 *)
%o A321616 (PARI) lista(nn) = {for (k=1, nn, if (isprime(p=k^2 + (k-1)^2) && isprime(k^p - (k-1)^p), print1(p, ", ")););} \\ _Michel Marcus_, Nov 18 2018
%Y A321616 Cf. A002144, A121091.
%Y A321616 Subsequence of A027862.
%K A321616 nonn,more
%O A321616 1,1
%A A321616 _Thomas Ordowski_, Nov 15 2018