A354805 - OEIS

%I #29 Jun 11 2022 11:45:01

%S 1,0,1,1,0,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,0,1,0,0,

%T 0,1,1,0,0,0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,

%U 0,1,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1

%N Triangle T(n,k) is an array of characteristic functions of primes involved in Goldbach's partitions of 2n.

%C The triangle T(n, k) consists of zeros and ones where row n is a sort of characteristic function of the primes used in Goldbach's partition of 2n into two odd primes by means of the indices k of the ones, these indices giving the deviation from n of the selected primes.

%C It can be observed that, in table T considered as a matrix, diagonals and antidiagonals starting at position (n,0) with n composite are all zeros.

%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>

%F Where T(n,k) is 1, the corresponding partition of 2n is (n-k, n+k).

%e Triangle begins:

%e   {1},

%e   {0, 1},

%e   {1, 0, 1},

%e   {0, 1, 0, 0},

%e   {1, 0, 0, 0, 1},

%e   {0, 0, 0, 1, 0, 1},

%e   {0, 0, 1, 0, 1, 0, 0},

%e   {0, 0, 0, 1, 0, 0, 0, 1},

%e   {1, 0, 0, 0, 0, 0, 1, 0, 1},

%e   ...

%e Example: row 11 is {1, 0, 0, 0, 0, 0, 1, 0, 1}, then, indices k of ones are 0, 6, 8, so, adding 11 gives back the primes 11, 17, 19 and (subtracting from 22) the partition {{11, 11}, {17, 5}, {19, 3}}.

%t nmin = 3; nmax = 16;

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

%t Table[T[n, k], {n, nmin, nmax}, {k, 0, n-nmin}] // Flatten (* _Jean-François Alcover_, Jun 11 2022 *)

%Y Cf. A002375 (row sums, the main entry for this sequence), A010051 (column k=0), A101264 (main diagonal T(n,n-3)).

%K nonn,tabl

%O 3

%A _Jean-François Alcover_, Jun 07 2022