A171637 Triangle read by rows in which row n lists the distinct primes of the distinct decompositions of 2n into unordered sums of two primes.

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

Offset: 2

Author

Juri-Stepan Gerasimov, Dec 13 2009

Keywords

• nonn
• tabf

Comments

Each entry of the n-th row is a prime p from the n-th row of A002260 such that 2n-p is also prime. If A002260 is read as the antidiagonals of a square array, this sequence can be read as an irregular square array (see example below). The n-th row has length A035026(n). This sequence is the nonzero subsequence of A154725. - Jason Kimberley, Jul 08 2012

Examples

			a(2)=2 because for row 2: 2*2=2+2; a(3)=3 because for row 3: 2*3=3+3; a(4)=3 and a(5)=5 because for row 4: 2*4=3+5; a(6)=3, a(7)=5 and a(8)=7 because for row 5: 2*5=3+7=5+5.
The table starts:
2;
3;
3,5;
3,5,7;
5,7;
3,7,11;
3,5,11,13;
5,7,11,13;
3,7,13,17;
3,5,11,17,19;
5,7,11,13,17,19;
3,7,13,19,23;
5,11,17,23;
7,11,13,17,19,23;
3,13,19,29;
3,5,11,17,23,29,31;
As an irregular square array [_Jason Kimberley_, Jul 08 2012]:
3 . 3 . 3 . . . 3 . 3 . . . 3 . 3
. . . . . . . . . . . . . . . .
5 . 5 . 5 . . . 5 . 5 . . . 5
. . . . . . . . . . . . . .
7 . 7 . 7 . . . 7 . 7 . .
. . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . .
11. 11. 11. . . 11
. . . . . . . .
13. 13. 13. .
. . . . . .
. . . . .
. . . .
17. 17
. .
19
		

Links

• T. D. Noe, Rows n = 2..250, flattened
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Related triangles: A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727, A184995. - Jason Kimberley, Sep 03 2011

Cf. A020481 (left edge), A020482 (right edge), A238778 (row sums), A238711 (row products), A000040, A010051.

Programs

Haskell

a171637 n k = a171637_tabf !! (n-2) !! (k-1)
a171637_tabf = map a171637_row [2..]
a171637_row n = reverse $ filter ((== 1) . a010051) $
   map (2 * n -) $ takeWhile (<= 2 * n) a000040_list
-- Reinhard Zumkeller, Mar 03 2014

Mathematica

Table[ps = Prime[Range[PrimePi[2*n]]]; Select[ps, MemberQ[ps, 2*n - #] &], {n, 2, 50}] (* T. D. Noe, Jan 27 2012 *)

Extensions

Keyword:tabl replaced by tabf, arbitrarily defined a(1) removed and entries checked by R. J. Mathar, May 22 2010

Definition clarified by N. J. A. Sloane, May 23 2010