A317745 Square array T(n,k) (n >= 1, k >= 1) read by antidiagonals: first row and column are A085090, other entries equal sum of entries in first row and first column.

0, 3, 3, 5, 6, 5, 7, 8, 8, 7, 0, 10, 10, 10, 0, 11, 3, 12, 12, 3, 11, 13, 14, 5, 14, 5, 14, 13, 0, 16, 16, 7, 7, 16, 16, 0, 17, 3, 18, 18, 0, 18, 18, 3, 17, 19, 20, 5, 20, 11, 11, 20, 5, 20, 19, 0, 22, 22, 7, 13, 22, 13, 7, 22, 22, 0, 23, 3, 24, 24, 0, 24, 24, 0, 24, 24, 3, 23

Offset: 1

Author

Fred Daniel Kline, Aug 05 2018

Keywords

• nonn
• tabl

Comments

This is related to Goldbach's conjecture, since entries for which the leftmost entry and the top entry are both nonzero are the sums of two primes.

The successive antidiagonals may also be regarded as the rows of a triangle, having A085090 as outside diagonals.

Examples

			Beginning of the array. All elements are equal to topmost value plus leftmost value.
   0  3  5  7  0 11 13  0 17 19  0 23
   3  6  8 10  3 14 16  3 20 22  3
   5  8 10 12  5 16 18  5 22 24
   7 10 12 14  7 18 20  7 24
   0  3  5  7  0 11 13  0
  11 14 16 18 11 22 24
  13 16 18 20 13 24
   0  3  5  7  0
  17 20 22 24
  19 22 24
   0  3
  23
		

Links

• Gustavo Funes, Damian Gulich, Leopoldo Garavaglia and Mario Garavaglia, Hidden Symmetries Among Primes, Form and Symmetry: Art and Science, Buenos Aires Congress, 2007, Section 4, Figure 10.
• Fred Daniel Kline, Goldbach Illustrated

Crossrefs

Cf. A085090.

Programs

Mathematica

i[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[i, 82];
r[k_] := Table[A085090[[j]] + A085090[[k - j + 1]], {j, 1, k}];
a = Array[r, 12] // Flatten,

PARI

a085090(n) = if (isprime(p=2*n-1), p, 0);
row(n) = vector(n, k, a085090(k) + a085090(n-k+1));
tabl(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Aug 09 2018

Formula

T(n, k) = A085090(n) + A085090(k).

Extensions

Edited by N. J. A. Sloane, Sep 09 2018