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A354836 Triangle T(n,k) where, if n-k and n+k are prime, T(n,k) = n+k is the greater term of a Goldbach partition of 2n into two odd primes, or zero otherwise.

%I A354836 #39 Mar 22 2025 19:04:35
%S A354836 3,0,5,5,0,7,0,7,0,0,7,0,0,0,11,0,0,0,11,0,13,0,0,11,0,13,0,0,0,0,0,
%T A354836 13,0,0,0,17,11,0,0,0,0,0,17,0,19,0,13,0,0,0,17,0,19,0,0,13,0,0,0,0,0,
%U A354836 19,0,0,0,23,0,0,0,17,0,0,0,0,0,23,0,0,0,0,17,0,19,0,0,0,23,0,0,0,0
%N A354836 Triangle T(n,k) where, if n-k and n+k are prime, T(n,k) = n+k is the greater term of a Goldbach partition of 2n into two odd primes, or zero otherwise.
%C A354836 This sequence has the same structure as A354805, which could be considered as sort of its characteristic function.
%H A354836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A354836 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s conjecture">Goldbach's conjecture</a>
%e A354836 Triangle begins:
%e A354836     3;
%e A354836     0, 5;
%e A354836     5, 0, 7;
%e A354836     0, 7, 0, 0;
%e A354836     7, 0, 0, 0,11;
%e A354836     0, 0, 0,11, 0,13;
%e A354836     0, 0,11, 0,13, 0, 0;
%e A354836     0, 0, 0,13, 0, 0, 0,17;
%e A354836    11, 0, 0, 0, 0, 0,17, 0,19;
%e A354836    ...
%e A354836 Example: for n=11, row {11,0,0,0,0,0,17,0,19}, when stripped of its zeros and subtracted from 2n=22, gives the partitions {{11,11},{17,5},{19,3}}.
%t A354836 nmin = 3; nmax = 16;
%t A354836 T[n_ /; n >= nmin, k_ /; k >= 0] := If[PrimeQ[n-k] && PrimeQ[n+k], n+k, 0];
%t A354836 Table[T[n, k], {n, nmin, nmax}, {k, 0, n - nmin}] // Flatten
%Y A354836 Cf. A085090 (main diagonal), A061397 (column k=0 prepended with (0,2)), A145091 (column k=1 prepended with (0,2,3,0)), A354805.
%K A354836 nonn,tabl
%O A354836 3,1
%A A354836 _Jean-François Alcover_, Jun 12 2022