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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.

%I A171637 #34 Aug 03 2014 14:01:25
%S A171637 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,
%T A171637 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,
%U A171637 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
%N 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.
%C A171637 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
%H A171637 T. D. Noe, <a href="/A171637/b171637.txt">Rows n = 2..250, flattened</a>
%H A171637 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A171637 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.
%e A171637 The table starts:
%e A171637 2;
%e A171637 3;
%e A171637 3,5;
%e A171637 3,5,7;
%e A171637 5,7;
%e A171637 3,7,11;
%e A171637 3,5,11,13;
%e A171637 5,7,11,13;
%e A171637 3,7,13,17;
%e A171637 3,5,11,17,19;
%e A171637 5,7,11,13,17,19;
%e A171637 3,7,13,19,23;
%e A171637 5,11,17,23;
%e A171637 7,11,13,17,19,23;
%e A171637 3,13,19,29;
%e A171637 3,5,11,17,23,29,31;
%e A171637 As an irregular square array [_Jason Kimberley_, Jul 08 2012]:
%e A171637 3 . 3 . 3 . . . 3 . 3 . . . 3 . 3
%e A171637 . . . . . . . . . . . . . . . .
%e A171637 5 . 5 . 5 . . . 5 . 5 . . . 5
%e A171637 . . . . . . . . . . . . . .
%e A171637 7 . 7 . 7 . . . 7 . 7 . .
%e A171637 . . . . . . . . . . . .
%e A171637 . . . . . . . . . . .
%e A171637 . . . . . . . . . .
%e A171637 11. 11. 11. . . 11
%e A171637 . . . . . . . .
%e A171637 13. 13. 13. .
%e A171637 . . . . . .
%e A171637 . . . . .
%e A171637 . . . .
%e A171637 17. 17
%e A171637 . .
%e A171637 19
%t A171637 Table[ps = Prime[Range[PrimePi[2*n]]]; Select[ps, MemberQ[ps, 2*n - #] &], {n, 2, 50}] (* _T. D. Noe_, Jan 27 2012 *)
%o A171637 (Haskell)
%o A171637 a171637 n k = a171637_tabf !! (n-2) !! (k-1)
%o A171637 a171637_tabf = map a171637_row [2..]
%o A171637 a171637_row n = reverse $ filter ((== 1) . a010051) $
%o A171637    map (2 * n -) $ takeWhile (<= 2 * n) a000040_list
%o A171637 -- _Reinhard Zumkeller_, Mar 03 2014
%Y A171637 Related triangles: A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727, A184995. - _Jason Kimberley_, Sep 03 2011
%Y A171637 Cf. A020481 (left edge), A020482 (right edge), A238778 (row sums), A238711 (row products), A000040, A010051.
%K A171637 nonn,tabf
%O A171637 2,1
%A A171637 _Juri-Stepan Gerasimov_, Dec 13 2009
%E A171637 Keyword:tabl replaced by tabf, arbitrarily defined a(1) removed and entries checked by _R. J. Mathar_, May 22 2010
%E A171637 Definition clarified by _N. J. A. Sloane_, May 23 2010