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%I #17 Oct 30 2023 15:47:42
%S 4,7,8,9,11,12,15,17,19,20,21,23,24,25,27,29,31,32,33,35,36,39,41,43,
%T 44,45,47,49,51,53,55,56,57,59,60,61,63,65,67,69,71,72,73,75,77,79,80,
%U 81,83,84,85,87,89,91,92,93,95,97,99,101,103,104,105,107
%N Numbers k such that s + t = k with 0 < s < t where t and t - s are both prime.
%e 4 is in the sequence since there are numbers, s=1 and t=3, that satisfy s + t = 4, where s < t, t = 3 (prime) and t - s = 3 - 1 = 2 (prime).
%e 7 is in the sequence since there are numbers, s=2 and t=5 that satisfy s + t = 7, where s < t, t = 5 (prime) and t - s = 5 - 2 = 3 (prime).
%t Flatten[Table[If[Sum[(PrimePi[n - i] - PrimePi[n - i - 1]) (PrimePi[n - 2 i] - PrimePi[n - 2 i - 1]), {i, Floor[n/2]}] > 0, n, {}], {n, 100}]]
%o (PARI) isok(k) = {forprime (t=1, k, if (((s = k - t) < t) && (s > 0) && isprime(t-s), return (1)););} \\ _Michel Marcus_, Jul 20 2019
%Y Cf. A309152, A309265.
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Jul 19 2019