A154727 Triangle read by rows in which row n lists all the pairs of prime numbers that are equidistant from n, or only n if there is no such pair, as shown below in the example.

1, 2, 3, 3, 5, 3, 7, 5, 7, 3, 11, 3, 5, 11, 13, 5, 7, 11, 13, 3, 7, 13, 17, 3, 5, 17, 19, 5, 7, 11, 13, 17, 19, 3, 7, 19, 23, 5, 11, 17, 23, 7, 11, 13, 17, 19, 23, 3, 13, 19, 29, 3, 5, 11, 23, 29, 31, 5, 7, 13, 17, 19, 23, 29, 31, 7, 31, 3, 11, 17

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 >= 4. - Nathaniel Johnston, Apr 18 2011

Examples

			Triangle begins:
                          1
                          2
                          3
                       3, .  5
                    3, .  .  .  7
                 .  .  5, .  7, . .
              3, .  .  .  .  .  .  . 11
           3, .  5, .  .  .  .  . 11, . 13
        .  .  5, .  7, .  .  . 11, . 13, .  .
     3, .  .  .  7, .  .  .  .  . 13, .  .  . 17
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Wolfram MathWorld, Goldbach Conjecture

Crossrefs

Cf. A000040, A154720, A154721, A154722, A154723, A154724, A154725, A154726.

Programs

Maple

print(1):print(2):print(3):for n from 1 to 15 do for k from 1 to 2*n-1 do if(not k=n and (isprime(k) and isprime(2*n-k)))then print(k):fi:od:od: # Nathaniel Johnston, Apr 18 2011

Mathematica

Table[n + Union@ Join[#, -#] /. {} -> {n} &@ Select[DeleteCases[n - Prime@ Range[2, PrimePi@ n], 0], AllTrue[n + # {-1, 1}, PrimeQ] &], {n, 20}] // Flatten (* Michael De Vlieger, Feb 03 2019 *)

Extensions

a(24)-a(70) from Nathaniel Johnston, Apr 18 2011