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A356478 a(n) is the least k such that there are exactly n primes p <= k such that 2*k-p and p*(2*k-p)+2*k are also prime.

%I A356478 #58 Sep 05 2022 09:10:42
%S A356478 2,4,11,15,21,35,42,111,81,117,126,60,291,147,225,417,210,330,357,555,
%T A356478 561,375,315,477,735,552,420,975,630,585,816,840,930,1925,1302,1170,
%U A356478 1140,2202,1215,1155,1911,1551,2031,1590,1365,2136,1425,2562,1740,1485,2331,2790,2160,2100,2640,2010,3681,2400,1785,2262,3252,2622,2940,1575,2310,2541,3987,2772
%N A356478 a(n) is the least k such that there are exactly n primes p <= k such that 2*k-p and p*(2*k-p)+2*k are also prime.
%C A356478 a(n) is the least k such that A356864(k) = n.
%H A356478 Robert Israel, <a href="/A356478/b356478.txt">Table of n, a(n) for n = 0..500</a>
%e A356478 a(3) = 15 because there are exactly 3 primes p <= 15 with 30-p and p*(30-p)+30 prime, namely 7, 11 and 13, and no smaller number works.
%p A356478 f:= proc(n) local p,q,t;
%p A356478   p:= 1: t:= 0:
%p A356478   do
%p A356478     p:= nextprime(p);
%p A356478     q:= n-p;
%p A356478     if q <= p then return t fi;
%p A356478     if isprime(q) and isprime(p*q+n) then t:= t+1 fi;
%p A356478   od
%p A356478 end proc:
%p A356478 V:= Array(0..100): V[0]:= 2: count:= 1:
%p A356478 for nn from 2 while count < 101 do
%p A356478   v:= f(2*nn);
%p A356478   if v > 100 then next fi;
%p A356478   if V[v] = 0 then count:= count+1; V[v]:= nn; fi;
%p A356478 od:
%p A356478 convert(V,list);
%Y A356478 Cf. A072511, A356864.
%K A356478 nonn
%O A356478 0,1
%A A356478 _J. M. Bergot_ and _Robert Israel_, Sep 01 2022