A234316 Irregular triangle T, read by rows, such that row n lists the larger parts of the Goldbach partitions of 2n (in decreasing order).

2, 3, 5, 7, 5, 7, 11, 7, 13, 11, 13, 11, 17, 13, 19, 17, 11, 19, 17, 13, 23, 19, 13, 23, 17, 23, 19, 17, 29, 19, 31, 29, 23, 17, 31, 29, 23, 19, 31, 19, 37, 29, 23, 37, 31, 29, 23, 41, 37, 31, 43, 41, 29, 23, 43, 41, 37, 31, 29, 47, 43, 37, 31, 47, 41, 29, 47, 43, 41, 37, 31

Offset

2,1

Comments

Row n has first entry A060308(n), and length A045917(n). If Goldbach's conjecture is true, then each row of the triangle contains at least 1 entry.

This is the companion irregular triangle to A184995. See the first formula. - Wolfdieter Lang, May 14 2016

Links

Table of n, a(n) for n=2..72.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Formula

T(n,i) = 2n - A184995(n,i).

T(n,i) = n + A182138(n,i). - Ralf Stephan, Dec 26 2013

Example

The irregular triangle T(n,i) begins:
   n | 2*n | i = 1   2   3   4   5   6 ...
  ---+-----+------------------------------
   2 |   4 |     2
   3 |   6 |     3
   4 |   8 |     5
   5 |  10 |     7   5
   6 |  12 |     7
   7 |  14 |    11   7
   8 |  16 |    13  11
   9 |  18 |    13  11
  10 |  20 |    17  13
  11 |  22 |    19  17  11
  12 |  24 |    19  17  13
  13 |  26 |    23  19  13
  14 |  28 |    23  17
  15 |  30 |    23  19  17
  16 |  32 |    29  19
  17 |  34 |    31  29  23  17
  18 |  36 |    31  29  23  19
  19 |  38 |    31  19
  20 |  40 |    37  29  23
  21 |  42 |    37  31  29  23
  22 |  44 |    41  37  31
  23 |  46 |    43  41  29  23
  24 |  48 |    43  41  37  31  29
  25 |  50 |    47  43  37  31
  26 |  52 |    47  41  29
  27 |  54 |    47  43  41  37  31
  28 |  56 |    53  43  37
  29 |  58 |    53  47  41  29
  30 |  60 |    53  47  43  41  37  31
 ... Reformatted and extended. - Wolfdieter Lang, May 14 2016

Mathematica

Table[First /@ DeleteDuplicates@ Map[Sort[{#, 2 n - #}, Greater] &, Select[2 n - Prime@ Range@ PrimePi[2 n], PrimeQ]], {n, 30}] // Flatten (* Michael De Vlieger, May 15 2016 *)

Prog

(PARI) for(n=2, 18, forprime(p=2, n, if(isprime(2*n-p), print1(2*n-p", ")))) \\ Ralf Stephan, Dec 26 2013

Crossrefs

Cf. A182138, A184995.

Sequence in context: A246258 A126048 A142349 * A284630 A345872 A081622

Adjacent sequences: A234313 A234314 A234315 * A234317 A234318 A234319

Keyword

nonn,tabf

Author

Wesley Ivan Hurt, Dec 23 2013

Status

approved