A230762 -id:A230762 - cp's OEIS frontend

A230822 a(n) is the smallest integer that makes A230762(n) the commonest number of decompositions of 2k into an unordered sum of two odd primes, where 3 <= k <= a(n).

3, 8, 28, 46, 47, 139, 146, 173, 262, 526, 857, 2029, 2239, 2251, 2659, 3184, 3592, 3793, 5209, 8777, 10124, 12872, 15439, 24979, 27241, 29314, 29416, 37652, 42589, 60524, 80449, 101704, 147304, 156841, 170899, 180046, 204916, 230149, 239048, 390826, 488647, 530609, 701497

Offset: 1

Lei Zhou, Oct 30 2013

If making a statistical bar chart with x-axis denoting the number of decompositions of an even number, and y-axis denoting the number of hits of an x-axis value for all 3 <= k <= m, there are one or more commonest x value presenting with highest y value.  Such commonest x values increase when m increases. a(n) is the smallest m value to make A230762(n) one of the commonest number or prime decomposition of 2k in the range of 3 <= k <= m.
Hypothesis: With the increase of m, the commonest number of decompositions of 2k into an unordered sum of two odd primes in the range of 3 <= k <= m ascends.
This hypothesis derives that a(n) is an ascending sequence.  Or say, a(n+1) > a(n).

			When m=3, k has only one value 3, 2k=6=3+3.  Only one possible decomposition, making a decomposition statistics {{x,y}}={{1,1}}.  So a(1)=3;
When m=4, k gets another value 4, 2k=8=3+5. The decomposition statistics {{x,y}}={{1,2}};...
Thereafter, k=5 makes 2k=10=5+5=3+7, {{x,y}}={{1,2},{2,1}}, the commonest value is still 1.
k=6, 2k=12=5+7, {{x,y}}={{1,3},{2,1}}, commonest x is still 1.
k=7, 2k=14=3+11=7+7, {{x,y}}={{1,3},{2,2}}, commonest x is still 1.
k=8, 2k=16=3+13=5+11, {{x,y}}={{1,3},{2,3}}, except for 1, 2 is now eligible to be the new possible commonest x, so a(2)=8 (the current k value).
...
Counting up to k=28, the decomposition statistics is {{1,3},{2,8},{3,8},{4,5},{5,2}}, 2 and 3 are now the commonest decompositions. It is the first time for 3 to appear.  So a(3)=28 (the current k value).

Lei Zhou, Table of n, a(n) for n = 1..45

Cf. A002375, A230762.

check = 0; ns = {}; mpos = 0; res = {}; sres = 0; s = {}; size = 0; k = 2;
While[k++; k2 = 2*k; p2 = k - 1; ct = 0;
  While[p2 = NextPrime[p2]; p2 < k2, p1 = k2 - p2;
   If[PrimeQ[p1], ct++]];
  (*Calculate Goldbach decomposition*)
  If[ct > size, Do[AppendTo[s, 0], {i, size + 1, ct}]; size = ct];
  (*and construct statistics in array s*)s[[ct]]++; m = Max[s];
  aa = Position[s, m]; la = Length[aa];
  Do[a = aa[[pos, 1]];
   If[a > sres, While[sres < a, AppendTo[res, 0]; sres++];
    res[[a]] = n; goal = Length[res];
    (*Generate list of n values where a new commonest appears*)
    If[mpos < goal, mpos = goal; check++; AppendTo[ns, k]]],
   (*Compose elements of this sequence into a list*)
   {pos, 1, la}];
  check < 16];
ns

Lei Zhou, Nov 08 2013, uploaded a b-file, extending the known elements to the 45th.

cp's OEIS Frontend

A230822 a(n) is the smallest integer that makes A230762(n) the commonest number of decompositions of 2k into an unordered sum of two odd primes, where 3 <= k <= a(n).

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