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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.

%I A154723 #19 Feb 16 2025 08:33:09
%S A154723 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,
%T A154723 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,
%U A154723 11,13,17,19,23,29,1,3,13,19,29,31,3
%N 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.
%C A154723 If the extended Goldbach conjecture is true, such a pair exists in row n for all n >= 2. - _Nathaniel Johnston_, Apr 18 2011
%H A154723 Nathaniel Johnston, <a href="/A154723/b154723.txt">Table of n, a(n) for n = 1..8000</a>
%H A154723 Abel Jansma, <a href="https://esc.fnwi.uva.nl/thesis/centraal/files/f1541951402.pdf">E_8 Symmetry Structures in the Ising model</a>, Master's thesis, University of Amsterdam, 2018.
%H A154723 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>
%e A154723 Triangle begins:
%e A154723                              1
%e A154723                           1,    3
%e A154723                        1,          5
%e A154723                     1,    3,    5,    7
%e A154723                        3,          7,
%e A154723               1,          5,    7,         11
%e A154723            1,    3,                     11,   13
%e A154723               3,    5,               11,   13,
%e A154723      1,          5,    7,         11,   13,         17
%e A154723   1,    3,          7,               13,         17,   19
%p A154723 isnotcomp:=proc(n)return (n=1 or isprime(n)) end:
%p A154723 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
%t A154723 Table[If[Length@ # == 1, #, DeleteCases[#, n]] &@ Union@ Flatten@ Select[IntegerPartitions[2 n, {2}], AllTrue[#, ! CompositeQ@ # &] &], {n, 17}] // Flatten (* _Michael De Vlieger_, Dec 06 2018 *)
%Y A154723 Cf. A000040, A008578, A154720, A154721, A154722, A154724, A154725, A154726, A154727.
%K A154723 easy,nonn,tabf
%O A154723 1,3
%A A154723 _Omar E. Pol_, Jan 14 2009, Jan 16 2009
%E A154723 a(36)-a(70) from _Nathaniel Johnston_, Apr 18 2011