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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)
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.

LINKS

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

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

Adjacent sequences:  A230759 A230760 A230761 * A230763 A230764 A230765

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

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