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A358573 a(n) = smallest prime p such that q, r and s are all prime, where q = p + 2*(2*n + 1), r = (p - 2*n - 1)/2, and s = (q + 2*n + 1)/2.

%I A358573 #15 Dec 31 2022 15:29:02
%S A358573 11,13,19,17,19,229,47,29,163,29,31,37,47,53,1231,41,43,61,83,61,439,
%T A358573 1217,59,73,59,61,67,89,83,541,71,73,103,593,271,349,83,89,103,461,
%U A358573 239,97,107,97,211,149,107,229,263,181,499,317,139,1453,131,809,127,137,163
%N A358573 a(n) = smallest prime p such that q, r and s are all prime, where q = p + 2*(2*n + 1), r = (p - 2*n - 1)/2, and s = (q + 2*n + 1)/2.
%C A358573 Equivalently, smallest prime of the form (p + q - 2*n - 1), where p is prime, q = p + 2*(2*n + 1) is prime, and (p + q + 2*n + 1) is also prime.
%C A358573 a(n) is the first term of the sequence of numbers m such that (m - 2*n - 2), (m - 1), (m + 4*n + 1) and (m + 6*n + 2) cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
%C A358573 Such sequence contains only prime numbers which are the lesser of a pair of primes (p, q) such that the pair (r, s) also forms a pair of primes with the same difference, where q = p + 2*(2*n + 1), r = (p - 2*n - 1)/2, and s = (q + 2*n + 1)/2.
%e A358573 229 is the lesser prime in the pair (229, 251) with difference 2*(2*5+1) = 22, and the couple (229-22/2)/2 = 109 and (251+22/2)/2 = 131 forms another prime pair with distance 22, and there is no prime lower than 229 with this property. Hence a(5) = 229.
%t A358573 a[n_] := Module[{p=2, q, r, s}, While[!AllTrue[{(q = p + 2*(2*n + 1)), (r = (p - 2*n - 1)/2), (s = (q + 2*n + 1)/2)}, #>0 && PrimeQ[#] &], p = NextPrime[p]]; p]; Array[a, 60, 0] (* _Amiram Eldar_, Nov 23 2022 *)
%o A358573 (PARI) a(n) = my(p=2, q); while(!isprime(q = p + 2*(2*n + 1)) || !isprime((p - 2*n - 1)/2) || !isprime((q + 2*n + 1)/2), p=nextprime(p+1)); p; \\ _Michel Marcus_, Nov 23 2022
%Y A358573 Cf. A001097, A001359, A006512, A077800, A158870.
%Y A358573 Cf. A023201, A255361, A254636, A256386.
%K A358573 nonn
%O A358573 0,1
%A A358573 _Lamine Ngom_, Nov 23 2022