Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #11 Oct 10 2024 08:18:42
%S 3,5,5,7,7,11,11,13,13,13,13,17,17,19,19,19,19,23,23,31,31,31,31,31,
%T 31,37,37,37,37,41,41,41,41,41,41,47,47,47,47,47,47,47,47,61,61,61,61,
%U 61,61,61,61,61,67,67,67,73,73,73,73,73,73,73,73,73,73,83
%N a(n) is the smallest prime number such that every even number from 6 to 2n can be written as the sum of two primes less than or equal to a(n).
%C The two primes stated in the name can be equal.
%H Lei Zhou, <a href="/A242189/b242189.txt">Table of n, a(n) for n = 3..5881</a>
%F a(n) = max_{3 <= i <= n} A234345(i). - _Robert Israel_, Oct 10 2024
%e n=3, 2*3=6=3+3. Since 3 is the smallest prime needed, a(3)=3.
%e n=4, 2*3=6=3+3, 2*4=8=5+3, Since 5 is the smallest prime needed, a(4)=5.
%e ...
%e n=14, we need to consider the even numbers from 6 to 2*14=28, while trying to minimize the larger prime number used to decompose such even numbers. 6=3+3; 8=5+3; 10=5+5; 12=7+5; 14=7+7; 16=11+5; 18=11+7; 20=13+7; 22=11+11; 24=13+11; 26=13+13; 28=17+11. The maximum prime number used is 17. So a(14)=17.
%p f:= proc(m) local p,p0;
%p p0:= m/2; if p0::even then p0:= p0+1 fi;
%p for p from p0 by 2 do if isprime(p) and isprime(m-p) then return p fi od
%p end proc:
%p R:= 3: m:= 3:
%p for i from 8 to 200 by 2 do
%p v:= f(i);
%p if v > m then R:= R,v; m:= v
%p else R:= R,m
%p fi
%p od:
%p R; # _Robert Israel_, Oct 10 2024
%t a = {2}; Table[found = 0; While[la = Length[a]; xx = 1; Do[yy = 0; Do[If[MemberQ[a, i*2 - a[[j]]], yy = 1], {j, 1, la}]; If[yy == 0, xx = 0], {i, 3, n}]; If[xx == 1, found = 1]; found == 0, AppendTo[a, NextPrime[Last[a]]]]; Last[a], {n, 3, 68}]
%Y Cf. A000040, A002375, A234345, A237628, A237638, A240708.
%K nonn
%O 3,1
%A _Lei Zhou_, May 06 2014
%E Name corrected by _Robert Israel_, Oct 10 2024