A317745 - OEIS

%I #72 Dec 05 2019 04:17:39

%S 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,

%T 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,

%U 19,0,22,22,7,13,22,13,7,22,22,0,23,3,24,24,0,24,24,0,24,24,3,23

%N 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.

%C 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.

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

%H Gustavo Funes, Damian Gulich, Leopoldo Garavaglia and Mario Garavaglia, <a href="http://www.mi.sanu.ac.rs/vismath/garavaglia2008/index.html">Hidden Symmetries Among Primes</a>, Form and Symmetry: Art and Science, Buenos Aires Congress, 2007, Section 4, Figure 10.

%H Fred Daniel Kline, <a href="/A317745/a317745_3.pdf">Goldbach Illustrated</a>

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

%e Beginning of the array. All elements are equal to topmost value plus leftmost value.

%e    0  3  5  7  0 11 13  0 17 19  0 23

%e    3  6  8 10  3 14 16  3 20 22  3

%e    5  8 10 12  5 16 18  5 22 24

%e    7 10 12 14  7 18 20  7 24

%e    0  3  5  7  0 11 13  0

%e   11 14 16 18 11 22 24

%e   13 16 18 20 13 24

%e    0  3  5  7  0

%e   17 20 22 24

%e   19 22 24

%e    0  3

%e   23

%t i[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[i, 82];

%t r[k_] := Table[A085090[[j]] + A085090[[k - j + 1]], {j, 1, k}];

%t a = Array[r, 12] // Flatten,

%o (PARI) a085090(n) = if (isprime(p=2*n-1), p, 0);

%o row(n) = vector(n, k, a085090(k) + a085090(n-k+1));

%o tabl(nn) = for (n=1, nn, print(row(n))); \\ _Michel Marcus_, Aug 09 2018

%Y Cf. A085090.

%K nonn,tabl

%O 1,2

%A _Fred Daniel Kline_, Aug 05 2018

%E Edited by _N. J. A. Sloane_, Sep 09 2018