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%I #32 Sep 22 2017 04:26:40
%S 2,5,3,11,7,13,23,17,29,19,47,37,59,41,31,97,79,127,83,67,43,197,163,
%T 257,167,137,89,53,397,331,521,337,277,179,107,61,797,673,1049,677,
%U 557,359,223,127,71,1597,1361,2099,1361,1117,719,449,257,149,73,3203
%N 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.
%C 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.
%C 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
%C The conjecture for column 1 is true iff A194598 and A193507 are equivalent. Is this the case? - _Bob Selcoe_, Oct 29 2015
%C 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
%e Northwest corner:
%e 2 5 11 23 47 97 197
%e 3 7 17 37 79 163 331
%e 13 29 59 127 257 521 1049
%e 19 41 83 167 337 677 1361
%e 31 67 137 277 557 1117 2237
%e 43 89 179 359 719 1439 2879
%e 53 107 223 449 907 1823 3659
%t seqL = 14; arr2[1] = {2}; Do[AppendTo[arr2[1], NextPrime[2*Last[arr2[1]]]], {seqL}];
%t 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 t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* _Peter J. C. Moses_, Sep 26 2013 *)
%Y Cf. A055496, A193507, A229607, A229609, A229610.
%Y Cf. A065545, A065546, A064934, A194598.
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Sep 26 2013
%E Incorrect comment deleted and example extended by _Peter Munn_, Jul 30 2017
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