A237628 -id:A237628 - cp's OEIS frontend

A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

1, 1, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 9, 11, 11, 11, 13, 16, 23, 25, 31, 47, 57, 63, 70, 74, 79, 82, 122, 131, 129, 180, 215, 219, 323, 367, 446, 501, 531, 661, 867, 897, 1311, 1471, 1691, 1695, 2130, 2288, 2833, 3363, 3891, 5435, 8068, 8867, 13476, 15451, 15897

Offset: 3

Lei Zhou, May 02 2014

			n=4, 2n=8. There is only one set of primes {3,5} such that 6=3+3, 8=3+5. So a(4)=1.
...
n=8, 2n=16. We can find two sets, {3,5,7,11} and {3,5,7,13} that have such features. So a(8)=2. Here any set with more primes either contains an unused prime number or one of these two sets is a subset of them, like {3,5,7,11,13}, and thus is not considered. So a(8)=2.
...
n=13, 2n=26. Five such sets are found: {3,5,7,11,13}, {3,5,7,13,17},{3,5,7,13,19}, {3,5,7,11,17,19}, {3,5,7,11,17,23}. So a(13)=5.

Lei Zhou, Table of n, a(n) for n = 3..79

Cf. A000040, A002375, A240708, A237628.

a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
  If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; Length[na], {n, 4, 60}](* Program lists the 4th item and beyond *)

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).

Original entry on oeis.org

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

Lei Zhou, May 06 2014

nonn

The two primes stated in the name can be equal.

			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.

Lei Zhou, Table of n, a(n) for n = 3..5881

Cf. A000040, A002375, A234345, A237628, A237638, A240708.

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

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}]

a(n) = max_{3 <= i <= n} A234345(i). - Robert Israel, Oct 10 2024

Name corrected by Robert Israel, Oct 10 2024

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A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

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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).

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