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

1,2

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.

Links

Table of n, a(n) for n=1..78.

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

Formula

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

Example

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

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,

Prog

(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

Crossrefs

Cf. A085090.

Sequence in context: A334075 A140351 A128444 * A204250 A131948 A134636

Adjacent sequences: A317742 A317743 A317744 * A317746 A317747 A317748

Keyword

nonn,tabl

Author

Fred Daniel Kline, Aug 05 2018

Extensions

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

Status

approved