login
A230822
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).
1
3, 8, 28, 46, 47, 139, 146, 173, 262, 526, 857, 2029, 2239, 2251, 2659, 3184, 3592, 3793, 5209, 8777, 10124, 12872, 15439, 24979, 27241, 29314, 29416, 37652, 42589, 60524, 80449, 101704, 147304, 156841, 170899, 180046, 204916, 230149, 239048, 390826, 488647, 530609, 701497
OFFSET
1,1
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. 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.
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.
This hypothesis derives that a(n) is an ascending sequence. Or say, a(n+1) > a(n).
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)=3;
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)=8 (the current k value).
...
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).
MATHEMATICA
check = 0; ns = {}; 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[ns, k]]],
(*Compose elements of this sequence into a list*)
{pos, 1, la}];
check < 16];
ns
CROSSREFS
Sequence in context: A005641 A377387 A260961 * A332727 A148845 A148846
KEYWORD
nonn,hard
AUTHOR
Lei Zhou, Oct 30 2013
EXTENSIONS
Lei Zhou, Nov 08 2013, uploaded a b-file, extending the known elements to the 45th.
STATUS
approved