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

Author

Lei Zhou, Oct 30 2013

Keywords

• nonn
• hard

Comments

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).

Examples

			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).
		

Links

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

Crossrefs

Cf. A002375, A230762.

Programs

Mathematica

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

Extensions

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