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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).
1
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
3,1
COMMENTS
The two primes stated in the name can be equal.
FORMULA
a(n) = max_{3 <= i <= n} A234345(i). - Robert Israel, Oct 10 2024
EXAMPLE
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.
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
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Lei Zhou, May 06 2014
EXTENSIONS
Name corrected by Robert Israel, Oct 10 2024
STATUS
approved