A247090 - OEIS

%I #20 Nov 28 2014 21:54:11

%S 1,2,1,1,1,1,2,1,1,1,1,3,1,1,1,2,3,1,1,1,1,1,1,1,5,1,1,1,2,1,1,5,3,1,

%T 1,1,1,1,1,5,3,1,1,1,1,2,1,1,5,3,1,1,1,1,1,1,3,1,1,3,1,1,1,1,1,1,2,3,

%U 1,1,1,1,1,1,1,1,1,1

%N Eric Rowland's generalization of A132199 as a rectangular array A read by upward antidiagonals.

%C Conjecture [Rowland] (paraphrased): Let A be the above array with entry A(n,k) in row n and column k. For each n, there exists an index N(n) >= 1 such that A(n,j) is either 1 or prime for all j > N(n).

%H Eric S. Rowland, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html">A natural prime-generating recurrence</a>, J. Integer Seq., 11 (2008), Article 08.2.8.

%e Array A begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...

%e 2, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...

%e 2, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...

%e 1, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...

%e 2, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...

%e 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...

%e 2, 1, 1, 1, 1, 7, 1, 1, 1,  1, 1, 13, ...

%e 1, 1, 1, 1, 1, 7, 1, 1, 1,  1, 1, 13, ...

%e 2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...

%e 1, 3, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...

%e 2, 1, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...

%e ...

%t (* Array A: *)

%t max := 13; b[n_, 1] := n; b[n_, k_] := b[n, k] = b[n, k - 1] + GCD[k, b[n, k - 1]]; Grid[Transpose[Differences[Transpose[Table[b[n, k], {n, max}, {k, max}]]]]]

%t (* Array antidiagonals flattened: *)

%t max := 13; b[n_, 1] := n; b[n_, k_] := b[n, k] = b[n, k - 1] + GCD[k, b[n, k - 1]]; Flatten[Table[Transpose[Differences[Transpose[Table[b[n, k], {n, max}, {k, max}]]]][[n - k + 1]][[k]], {n, max - 1}, {k, n}]]

%Y Cf. A106108, A132199, A137613.

%K nonn,tabl

%O 1,2

%A _L. Edson Jeffery_, Nov 18 2014