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

Author

Wesley Ivan Hurt, Dec 23 2013

Keywords

• nonn
• tabf

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

Examples

			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
		

Links

• 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

Crossrefs

Cf. A182138, A184995.

Programs

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 *)

PARI

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

Formula

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

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