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A230762 List of commonest number of decompositions of 2k into an unordered sum of two odd primes in range 3 <= k <= m, integer m >= 3, where m is explained below.

%I A230762 #20 Nov 08 2013 13:42:43
%S A230762 1,2,3,4,5,7,8,9,11,18,27,44,48,52,58,61,75,77,98,141,165,200,231,337,
%T A230762 360,378,384,466,517,697,880,1061,1400,1503,1615,1700,1896,2082,2163,
%U A230762 3242,3929,4232,5373
%N A230762 List of commonest number of decompositions of 2k into an unordered sum of two odd primes in range 3 <= k <= m, integer m >= 3, where m is explained below.
%C A230762 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, and fall on the x values listed in this sequence.
%C A230762 Hypothesis: With the increase of m, the commonest number of decompositions of 2n into an unordered sum of two odd primes in the range of 3 <= k <= m ascends.
%C A230762 This hypothesis derives that the corresponding smallest m to the terms of this sequence makes an ascending sequence.  Or say, when testing with m ascending, once a number a(n) enters this sequence, no number smaller than a(n) will be able to enter this sequence if they had not enter previous.
%H A230762 Lei Zhou, <a href="/A230762/b230762.txt">Table of n, a(n) for n = 1..45</a>
%e A230762 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)=1;
%e A230762 When m=4, k gets another value 4, 2k=8=3+5. The decomposition statistics {{x,y}}={{1,2}};...
%e A230762 Thereafter, k=5 makes 2k=10=5+5=3+7, {{x,y}}={{1,2},{2,1}}, the commonest value is still 1.
%e A230762 k=6, 2k=12=5+7, {{x,y}}={{1,3},{2,1}}, commonest x is still 1.
%e A230762 k=7, 2k=14=3+11=7+7, {{x,y}}={{1,3},{2,2}}, commonest x is still 1.
%e A230762 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)=2.
%e A230762 ...
%e A230762 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)=3.
%t A230762 check=0;posts={};mpos=0;res={};sres=0;s={};size=0;k=2;
%t A230762 While[k++;k2=2*k;p2=k-1;ct=0;
%t A230762   While[p2=NextPrime[p2];p2<k2,p1=k2-p2;If[PrimeQ[p1],ct++]];
%t A230762   (*Calculate Goldbach decomposition*)
%t A230762   If[ct>size, Do[AppendTo[s,0],{i,size+1,ct}]; size=ct];
%t A230762   (*and construct statistics in array s*)
%t A230762   s[[ct]]++;m=Max[s];aa=Position[s,m];la=Length[aa];
%t A230762   Do[a=aa[[pos,1]];
%t A230762     If[a>sres,
%t A230762       While[sres<a,AppendTo[res,0];sres++];res[[a]]=n;goal=Length[res];
%t A230762       (*Generate list of n values where a new commonest appears*)
%t A230762       If[mpos<goal,mpos=goal;check++;AppendTo[posts,mpos]]],
%t A230762       (*Compose elements of this sequence into a list*)
%t A230762     {pos,1,la}];
%t A230762   check<16];
%t A230762 posts
%Y A230762 Cf. A002375.
%K A230762 nonn,hard
%O A230762 1,2
%A A230762 _Lei Zhou_, Oct 29 2013
%E A230762 _Lei Zhou_, Nov 08 2013, uploaded a b-file extending the known elements of this sequence to the 45th.