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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 (list; graph; refs; listen; history; text; internal format)



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.


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


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.


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]];


      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}];




Cf. A002375.

Sequence in context: A192137 A192139 A080664 * A098776 A098897 A265641

Adjacent sequences:  A230759 A230760 A230761 * A230763 A230764 A230765




Lei Zhou, Oct 29 2013


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



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Last modified November 21 16:04 EST 2019. Contains 329371 sequences. (Running on oeis4.)