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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.
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%I #35 Jun 14 2022 01:41:43

%S 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 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 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 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 This sequence has the same structure as A354805, which could be considered as sort of its characteristic function.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s conjecture">Goldbach's conjecture</a>

%e Triangle begins:

%e 3;

%e 0, 5;

%e 5, 0, 7;

%e 0, 7, 0, 0;

%e 7, 0, 0, 0,11;

%e 0, 0, 0,11, 0,13;

%e 0, 0,11, 0,13, 0, 0;

%e 0, 0, 0,13, 0, 0, 0,17;

%e 11, 0, 0, 0, 0, 0,17, 0,19;

%e ...

%e 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 nmin = 3; nmax = 16;

%t T[n_ /; n >= nmin, k_ /; k >= 0] := If[PrimeQ[n-k] && PrimeQ[n+k], n+k, 0];

%t Table[T[n, k], {n, nmin, nmax}, {k, 0, n - nmin}] // Flatten

%Y Cf. A085090 (main diagonal), A061397 (column k=0 prepended with (0,2)), A145091 (column k=1 prepended with (0,2,3,0), A354805.

%K nonn,tabl

%O 3,1

%A _Jean-François Alcover_, Jun 12 2022