A154723 Triangle read by rows in which row n lists all the pairs of noncomposite numbers that are equidistant from n, or only n if there are no such pairs, as shown below in the example.

1, 1, 3, 1, 5, 1, 3, 5, 7, 3, 7, 1, 5, 7, 11, 1, 3, 11, 13, 3, 5, 11, 13, 1, 5, 7, 11, 13, 17, 1, 3, 7, 13, 17, 19, 3, 5, 17, 19, 1, 5, 7, 11, 13, 17, 19, 23, 3, 7, 19, 23, 5, 11, 17, 23, 1, 7, 11, 13, 17, 19, 23, 29, 1, 3, 13, 19, 29, 31, 3

Offset: 1

Author

Omar E. Pol, Jan 14 2009, Jan 16 2009

Keywords

• easy
• nonn
• tabf

Comments

If the extended Goldbach conjecture is true, such a pair exists in row n for all n >= 2. - Nathaniel Johnston, Apr 18 2011

Examples

			Triangle begins:
                             1
                          1,    3
                       1,          5
                    1,    3,    5,    7
                       3,          7,
              1,          5,    7,         11
           1,    3,                     11,   13
              3,    5,               11,   13,
     1,          5,    7,         11,   13,         17
  1,    3,          7,               13,         17,   19
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..8000
• Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.
• Wolfram MathWorld, Goldbach Conjecture

Crossrefs

Cf. A000040, A008578, A154720, A154721, A154722, A154724, A154725, A154726, A154727.

Programs

Maple

isnotcomp:=proc(n)return (n=1 or isprime(n)) end:
print(1):for n from 1 to 10 do for k from 1 to 2*n-1 do if(not k=n and (isnotcomp(k) and isnotcomp(2*n-k)))then print(k):fi:od:od: # Nathaniel Johnston, Apr 18 2011

Mathematica

Table[If[Length@ # == 1, #, DeleteCases[#, n]] &@ Union@ Flatten@ Select[IntegerPartitions[2 n, {2}], AllTrue[#, ! CompositeQ@ # &] &], {n, 17}] // Flatten (* Michael De Vlieger, Dec 06 2018 *)

Extensions

a(36)-a(70) from Nathaniel Johnston, Apr 18 2011