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 A341861 #9 Feb 22 2021 11:17:35
%S A341861 1,1,2,3,4,3,2,4,4,3,4,5,7,8,5,8,7,9,9,10,11,11,12,12,14,13,13,12,15,
%T A341861 14,14,17,15,19,18,12,19,13,19,20,22,20,24,21,15,21,21,23,25,26,23,26,
%U A341861 26,19,23,27,24,29,27,26,28,31,29,30,25,30,30,34,31,29,35
%N A341861 Number of primes among the (p-1)/2 numbers {2*p+1, 4*p+1, ..., (p-1)*p+1}, p = prime(n).
%C A341861 By Dirichlet's theorem on arithmetic progressions, we know there exists a prime q of the form 2*k*p+1. But the theorem does not give us any information about the size of the smallest q. It is conjectured that q < p^2. Moreover, it seems that a(n) goes to infinity as n increases.
%H A341861 Jianing Song, <a href="/A341861/b341861.txt">Table of n, a(n) for n = 2..10000</a>
%e A341861 Let P denote the set of prime numbers. Then:
%e A341861 a(8) = #({39, 77, 115, 153, 191, 229, 267, 305, 343} intersect P) = #{191, 229} = 2.
%e A341861 a(11) = #({63, 125, 187, 249, 311, 373, 435, 497, 559, 621, 683, 745, 807, 869, 931} intersect P) = #{311, 373, 683} = 3.
%o A341861 (PARI) a(n) = my(p=prime(n)); sum(k=1, (p-1)/2, isprime(2*k*p+1))
%Y A341861 Cf. A070846.
%K A341861 nonn
%O A341861 2,3
%A A341861 _Jianing Song_, Feb 21 2021