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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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%I #12 Nov 08 2013 13:42:54

%S 3,8,28,46,47,139,146,173,262,526,857,2029,2239,2251,2659,3184,3592,

%T 3793,5209,8777,10124,12872,15439,24979,27241,29314,29416,37652,42589,

%U 60524,80449,101704,147304,156841,170899,180046,204916,230149,239048,390826,488647,530609,701497

%N 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 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 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 This hypothesis derives that a(n) is an ascending sequence. Or say, a(n+1) > a(n).

%H Lei Zhou, <a href="/A230822/b230822.txt">Table of n, a(n) for n = 1..45</a>

%e 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 When m=4, k gets another value 4, 2k=8=3+5. The decomposition statistics {{x,y}}={{1,2}};...

%e Thereafter, k=5 makes 2k=10=5+5=3+7, {{x,y}}={{1,2},{2,1}}, the commonest value is still 1.

%e k=6, 2k=12=5+7, {{x,y}}={{1,3},{2,1}}, commonest x is still 1.

%e k=7, 2k=14=3+11=7+7, {{x,y}}={{1,3},{2,2}}, commonest x is still 1.

%e 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 ...

%e 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 check = 0; ns = {}; mpos = 0; res = {}; sres = 0; s = {}; size = 0; k = 2;

%t While[k++; k2 = 2*k; p2 = k - 1; ct = 0;

%t While[p2 = NextPrime[p2]; p2 < k2, p1 = k2 - p2;

%t If[PrimeQ[p1], ct++]];

%t (*Calculate Goldbach decomposition*)

%t If[ct > size, Do[AppendTo[s, 0], {i, size + 1, ct}]; size = ct];

%t (*and construct statistics in array s*)s[[ct]]++; m = Max[s];

%t aa = Position[s, m]; la = Length[aa];

%t Do[a = aa[[pos, 1]];

%t If[a > sres, While[sres < a, AppendTo[res, 0]; sres++];

%t res[[a]] = n; goal = Length[res];

%t (*Generate list of n values where a new commonest appears*)

%t If[mpos < goal, mpos = goal; check++; AppendTo[ns, k]]],

%t (*Compose elements of this sequence into a list*)

%t {pos, 1, la}];

%t check < 16];

%t ns

%Y Cf. A002375, A230762.

%K nonn,hard

%O 1,1

%A _Lei Zhou_, Oct 30 2013

%E _Lei Zhou_, Nov 08 2013, uploaded a b-file, extending the known elements to the 45th.