A229607 Square array read by antidiagonals downwards in which each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 2*p.

2, 3, 11, 5, 19, 17, 7, 37, 31, 29, 13, 73, 61, 53, 41, 23, 139, 113, 103, 79, 47, 43, 277, 223, 199, 157, 89, 59, 83, 547, 443, 397, 313, 173, 113, 67, 163, 1093, 883, 787, 619, 337, 223, 131, 71, 317, 2179, 1759, 1571, 1237, 673, 443, 257, 139, 97, 631

Offset: 1

Author

Clark Kimberling, Sep 26 2013

Keywords

• nonn
• tabl

Comments

Conjectures: (row 1) = A006992, (column 1) = A104272, and for each row r(k), the limit of r(k)/2^k exists. For rows 1 to 4, the respective limits are 0.303976..., 4.249137..., 6.857407..., 12.235210... .

From Pontus von Brömssen, Jan 18 2025: (Start)

Regarding the conjectures above:

- Row 1 is A006992 by definition.

- Column 1 is A164368, not A104272. It seems that the first column would be A104272 if no duplicates were allowed, i.e., if the prime p in a row were followed by the largest prime < 2*p not in a previous row; see A380277.

- The existence of the limits should follow from a strong version of Bertrand's postulate. For row 1, see formula in A006992.

(End)

Examples

			Northwest corner:
   2,    3,    5,    7,   13,   23,   43,   83, ...
  11,   19,   37,   73,  139,  277,  547, 1093, ...
  17,   31,   61,  113,  223,  443,  883, 1759, ...
  29,   53,  103,  199,  397,  787, 1571, 3137, ...
  41,   79,  157,  313,  619, 1237, 2473, 4943, ...
  47,   89,  173,  337,  673, 1327, 2647, 5281, ...
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)
• Wikipedia, Bertrand's postulate.

Crossrefs

Cf. A006992, A104272, A164368, A229608, A229609, A229610, A380277.

Programs

Mathematica

seqL = 14; arr1[1] = {2}; Do[AppendTo[arr1[1], NextPrime[2*Last[arr1[1]], -1]], {seqL}]; Do[tmp = Union[Flatten[Map[arr1, Range[z]]]]; arr1[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr1[z], NextPrime[2*Last[arr1[z]], -1]], {seqL}], {z, 2, 12}]; m = Map[arr1, 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 by Peter Munn, Aug 15 2017