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

2,1

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

Links

T. D. Noe, Rows n = 2..250, flattened

Index entries for sequences related to Goldbach conjecture

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A076368 A279931 A071049 * A140187 A214127 A381934

Adjacent sequences: A171634 A171635 A171636 * A171638 A171639 A171640

Keyword

nonn,tabf

Author

Juri-Stepan Gerasimov, Dec 13 2009

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

Status

approved