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 A348153 #32 Oct 07 2021 13:57:16
%S A348153 2,3,5,17,41,73,89,97,137,193,233,257,313,353,449,457,569,641,809,857,
%T A348153 929,1033,1049,1097,1129,1153,1193,1217,1289,1409,1433,1601,1609,1697,
%U A348153 1753,1889,1913,1993,2081,2137,2153,2273,2297,2393,2473,2617,2633,2657,2689,2713,2753,2777,2969
%N A348153 Primes for which there is no pair (k,q) with k a positive integer and q another prime, such that p=q*(2k+1)-2k.
%C A348153 There are primes p for which there exist a positive integer k and another prime q such that p=q*(2k+1)-2k. See A136020, A091180, A136061 and the subsequent sequences. Such k is called an "order" of the prime p. Note that q is necessarily larger than 2 and that 4*k is necessarily smaller than p-1. A prime may belong to more than one order, but the primes listed in the present sequence do not belong to any order.
%C A348153 As soon as they are larger than 8, all members minus 1 are multiples of 8.
%t A348153 lim = 2000; p = 2; listc = {}; listp = {}; While[p < lim, n = 1;
%t A348153 While[n <= (p - 3)/4,
%t A348153 If[PrimeQ[(p + 2 n)/(2 n + 1)], n = 2*p, n = n + 1]];
%t A348153 If[n == 2*p, AppendTo[listc, p]]; AppendTo[listp, p];
%t A348153 p = NextPrime[p]]; Complement[listp, listc]
%o A348153 (PARI) isok(p) = {if (isprime(p), for (k=1, (p-3)/4, my(q = (p+2*k)/(2*k+1)); if ((denominator(q)==1) && isprime(q), return(0));); return (1););} \\ _Michel Marcus_, Oct 07 2021
%Y A348153 Cf. A136020, A091180, A136061.
%K A348153 nonn
%O A348153 1,1
%A A348153 _René Gy_, Oct 03 2021
%E A348153 More terms from _Michel Marcus_, Oct 04 2021