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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).

%I A230822 #12 Nov 08 2013 13:42:54
%S A230822 3,8,28,46,47,139,146,173,262,526,857,2029,2239,2251,2659,3184,3592,
%T A230822 3793,5209,8777,10124,12872,15439,24979,27241,29314,29416,37652,42589,
%U A230822 60524,80449,101704,147304,156841,170899,180046,204916,230149,239048,390826,488647,530609,701497
%N 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).
%C A230822 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.
%C A230822 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.
%C A230822 This hypothesis derives that a(n) is an ascending sequence.  Or say, a(n+1) > a(n).
%H A230822 Lei Zhou, <a href="/A230822/b230822.txt">Table of n, a(n) for n = 1..45</a>
%e A230822 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;
%e A230822 When m=4, k gets another value 4, 2k=8=3+5. The decomposition statistics {{x,y}}={{1,2}};...
%e A230822 Thereafter, k=5 makes 2k=10=5+5=3+7, {{x,y}}={{1,2},{2,1}}, the commonest value is still 1.
%e A230822 k=6, 2k=12=5+7, {{x,y}}={{1,3},{2,1}}, commonest x is still 1.
%e A230822 k=7, 2k=14=3+11=7+7, {{x,y}}={{1,3},{2,2}}, commonest x is still 1.
%e A230822 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).
%e A230822 ...
%e A230822 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).
%t A230822 check = 0; ns = {}; mpos = 0; res = {}; sres = 0; s = {}; size = 0; k = 2;
%t A230822 While[k++; k2 = 2*k; p2 = k - 1; ct = 0;
%t A230822   While[p2 = NextPrime[p2]; p2 < k2, p1 = k2 - p2;
%t A230822    If[PrimeQ[p1], ct++]];
%t A230822   (*Calculate Goldbach decomposition*)
%t A230822   If[ct > size, Do[AppendTo[s, 0], {i, size + 1, ct}]; size = ct];
%t A230822   (*and construct statistics in array s*)s[[ct]]++; m = Max[s];
%t A230822   aa = Position[s, m]; la = Length[aa];
%t A230822   Do[a = aa[[pos, 1]];
%t A230822    If[a > sres, While[sres < a, AppendTo[res, 0]; sres++];
%t A230822     res[[a]] = n; goal = Length[res];
%t A230822     (*Generate list of n values where a new commonest appears*)
%t A230822     If[mpos < goal, mpos = goal; check++; AppendTo[ns, k]]],
%t A230822    (*Compose elements of this sequence into a list*)
%t A230822    {pos, 1, la}];
%t A230822   check < 16];
%t A230822 ns
%Y A230822 Cf. A002375, A230762.
%K A230822 nonn,hard
%O A230822 1,1
%A A230822 _Lei Zhou_, Oct 30 2013
%E A230822 _Lei Zhou_, Nov 08 2013, uploaded a b-file, extending the known elements to the 45th.