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 A356567 #38 Oct 05 2022 04:52:39
%S A356567 1,2,11,29,326
%N A356567 Numbers that generate increasing numbers of consecutive primes when doubled and added to the sequence of odd squares. (Positions of records in A354499.)
%C A356567 With a(1)=1, a(n) such that (2*x-1)^2 + 2*a(n) gives prime numbers for x=1 to k where the k for a(n) exceeds the k for a(n-1), a(n-2), ..., a(1).
%C A356567 Conjecture: this list is complete, since primes get farther apart as numbers increase. (2x-1)^2 + 2*29 generates many primes, with 38 of the first 43 and 105 of the first 156 values of x generating primes.
%C A356567 For the (2x-1)^2 + 2*29 values that are not prime, there seems to be a restriction on the factors. No values with prime factors below 29 were seen, nor were 41, 43, 53, 71, 73, 89, 97, ... For each of the other a(n) (or indeed any other natural number K), it seems there is a list of acceptable prime factors for the (2x-1)^2 + 2*K value. This gives a curious connection between addition and prime factors.
%C A356567 a(6) > 10^8, if it exists. - _Amiram Eldar_, Aug 15 2022
%e A356567 a(1)=1, because 1^2+2*1=3 and 3^2+2*1=11 are prime but 5^2+2*1=27 is not, and thus k=2.
%e A356567 a(2)=2, because 1^2+2*2=5 ... 7^2+2*2=53 are prime but 9^2+2*2=85 is not, thus k=4.
%e A356567 a(3)=11, because 1^2+2*11=23 ... 9^2+2*11=103 are prime, thus k=5.
%e A356567 a(4)=29, because 1^2+2*29=59 ... 27^2+2*29=787 are all prime, thus k=14.
%e A356567 a(5)=326 because 1^2+2*326=653 ... 35^2+2*352=1877 are all prime, thus k=18.
%t A356567 f[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; s = {}; fm = 0; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^3}]; s (* _Amiram Eldar_, Aug 15 2022 *)
%o A356567 (PARI) f(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ A354499
%o A356567 lista(nn) = my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", "));); \\ _Michel Marcus_, Aug 16 2022
%Y A356567 Cf. A016754 (odd squares), A354499 (consecutive primes generated by adding 2n to odd squares).
%Y A356567 Cf. A145202 and A188459 (related to last term).
%K A356567 nonn,more
%O A356567 1,2
%A A356567 _Steven M. Altschuld_, Aug 12 2022