A354805 Triangle T(n,k) is an array of characteristic functions of primes involved in Goldbach's partitions of 2n.

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, 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, 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

Offset

3

Comments

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.

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.

Links

Table of n, a(n) for n=3..107.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Formula

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

Example

Triangle begins:
  {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, 0, 1},
  {1, 0, 0, 0, 0, 0, 1, 0, 1},
  ...
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}}.

Mathematica

nmin = 3; nmax = 16;
T[n_ /; n >= nmin, k_ /; k >= 0] := Boole[PrimeQ[n-k] && PrimeQ[n+k]];
Table[T[n, k], {n, nmin, nmax}, {k, 0, n-nmin}] // Flatten (* Jean-François Alcover, Jun 11 2022 *)

Crossrefs

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

Sequence in context: A374646 A189295 A004539 * A294878 A023960 A129686

Adjacent sequences: A354802 A354803 A354804 * A354806 A354807 A354808

Keyword

nonn,tabl

Author

Jean-François Alcover, Jun 07 2022

Status

approved