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
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
KEYWORD
nonn,tabl
AUTHOR
Jean-François Alcover, Jun 07 2022
STATUS
approved