A242189 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).
3, 5, 5, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 47, 47, 61, 61, 61, 61, 61, 61, 61, 61, 61, 67, 67, 67, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 83
Offset: 3
Keywords
Examples
n=3, 2*3=6=3+3. Since 3 is the smallest prime needed, a(3)=3. n=4, 2*3=6=3+3, 2*4=8=5+3, Since 5 is the smallest prime needed, a(4)=5. ... 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.
Links
- Lei Zhou, Table of n, a(n) for n = 3..5881
Programs
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Maple
f:= proc(m) local p,p0; p0:= m/2; if p0::even then p0:= p0+1 fi; for p from p0 by 2 do if isprime(p) and isprime(m-p) then return p fi od end proc: R:= 3: m:= 3: for i from 8 to 200 by 2 do v:= f(i); if v > m then R:= R,v; m:= v else R:= R,m fi od: R; # Robert Israel, Oct 10 2024
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Mathematica
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}]
Formula
a(n) = max_{3 <= i <= n} A234345(i). - Robert Israel, Oct 10 2024
Extensions
Name corrected by Robert Israel, Oct 10 2024
Comments