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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.
2
1, 2, 3, 4, 5, 7, 8, 9, 11, 18, 27, 44, 48, 52, 58, 61, 75, 77, 98, 141, 165, 200, 231, 337, 360, 378, 384, 466, 517, 697, 880, 1061, 1400, 1503, 1615, 1700, 1896, 2082, 2163, 3242, 3929, 4232, 5373
OFFSET
1,2
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, and fall on the x values listed in this sequence.
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.
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.
EXAMPLE
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;
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)=2.
...
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.
MATHEMATICA
check=0; posts={}; 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[posts, mpos]]],
(*Compose elements of this sequence into a list*)
{pos, 1, la}];
check<16];
posts
CROSSREFS
Cf. A002375.
Sequence in context: A192137 A192139 A080664 * A098776 A098897 A265641
KEYWORD
nonn,hard
AUTHOR
Lei Zhou, Oct 29 2013
EXTENSIONS
Lei Zhou, Nov 08 2013, uploaded a b-file extending the known elements of this sequence to the 45th.
STATUS
approved