A229608 Square array read by antidiagonals downwards: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the least prime > 2*p.

2, 5, 3, 11, 7, 13, 23, 17, 29, 19, 47, 37, 59, 41, 31, 97, 79, 127, 83, 67, 43, 197, 163, 257, 167, 137, 89, 53, 397, 331, 521, 337, 277, 179, 107, 61, 797, 673, 1049, 677, 557, 359, 223, 127, 71, 1597, 1361, 2099, 1361, 1117, 719, 449, 257, 149, 73, 3203

Offset: 1

Author

Clark Kimberling, Sep 26 2013

Keywords

• nonn
• tabl

Comments

Conjectures: (row 1) = A055496, (column 1) = A193507, and for each row r(k), the limit of r(k)/2^k exists. For rows 1 to 4, the respective limits are 1.569985..., 2.677285..., 8.230592..., 10.709142...; see Franklin T. Adams-Watters's comment at A055496.

The above conjecture row 1 = A055496 is true; additionally, row 2 = A065545; row 3 = A065546; the first 5 terms of row 6 are a contiguous subsequence of A064934; and column 1 = A194598. - Bob Selcoe, Oct 27 2015; corrected by Peter Munn, Jul 30 2017

The conjecture for column 1 is true iff A194598 and A193507 are equivalent. Is this the case? - Bob Selcoe, Oct 29 2015

Column 1 diverges from A193507 at A(14,1) = 113, a prime not in A193507. 113 is in column 1 as it does not follow a prime in a row: 107 follows 53 and 127 follows 59, the next prime after 53. - Peter Munn, Jul 30 2017

Examples

			Northwest corner:
    2    5   11   23   47   97  197
    3    7   17   37   79  163  331
   13   29   59  127  257  521 1049
   19   41   83  167  337  677 1361
   31   67  137  277  557 1117 2237
   43   89  179  359  719 1439 2879
   53  107  223  449  907 1823 3659
		

Crossrefs

Cf. A055496, A193507, A229607, A229609, A229610.

Cf. A065545, A065546, A064934, A194598.

Programs

Mathematica

seqL = 14; arr2[1] = {2}; Do[AppendTo[arr2[1], NextPrime[2*Last[arr2[1]]]], {seqL}];
Do[tmp = Union[Flatten[Map[arr2, Range[z]]]]; arr2[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr2[z], NextPrime[2*Last[arr2[z]]]], {seqL}], {z, 2, 12}]; m = Map[arr2, Range[12]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Extensions

Incorrect comment deleted and example extended by Peter Munn, Jul 30 2017