This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A229607 #31 Jan 20 2025 22:53:41
%S A229607 2,3,11,5,19,17,7,37,31,29,13,73,61,53,41,23,139,113,103,79,47,43,277,
%T A229607 223,199,157,89,59,83,547,443,397,313,173,113,67,163,1093,883,787,619,
%U A229607 337,223,131,71,317,2179,1759,1571,1237,673,443,257,139,97,631
%N 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.
%C A229607 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... .
%C A229607 From _Pontus von Brömssen_, Jan 18 2025: (Start)
%C A229607 Regarding the conjectures above:
%C A229607 - Row 1 is A006992 by definition.
%C A229607 - 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.
%C A229607 - The existence of the limits should follow from a strong version of Bertrand's postulate. For row 1, see formula in A006992.
%C A229607 (End)
%H A229607 Pontus von Brömssen, <a href="/A229607/b229607.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals)
%H A229607 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bertrand's_postulate#Better_results">Bertrand's postulate</a>.
%e A229607 Northwest corner:
%e A229607 2, 3, 5, 7, 13, 23, 43, 83, ...
%e A229607 11, 19, 37, 73, 139, 277, 547, 1093, ...
%e A229607 17, 31, 61, 113, 223, 443, 883, 1759, ...
%e A229607 29, 53, 103, 199, 397, 787, 1571, 3137, ...
%e A229607 41, 79, 157, 313, 619, 1237, 2473, 4943, ...
%e A229607 47, 89, 173, 337, 673, 1327, 2647, 5281, ...
%t A229607 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 A229607 t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* _Peter J. C. Moses_, Sep 26 2013 *)
%Y A229607 Cf. A006992, A104272, A164368, A229608, A229609, A229610, A380277.
%K A229607 nonn,tabl
%O A229607 1,1
%A A229607 _Clark Kimberling_, Sep 26 2013
%E A229607 Incorrect comment deleted by _Peter Munn_, Aug 15 2017