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A271363 Irregular triangle read by rows: T(0, 0) = 2; T(i, j) is the j-th term in the least maximal chain of composites that is longer than the (i-1)-st least maximal chain of composites, where i>0.

%I A271363 #8 Apr 07 2016 02:58:37
%S A271363 2,4,3,14,15,17,18,21,25,31,40,55,77,111,163,50,69,99,147,225,353,60,
%T A271363 85,123,185,285,447,721,1185,1981,3363,5777,10039,82,119,177,273,429,
%U A271363 693,1135,1891,3201,5497,9543,16723,29579,52737,94705,171147,311101
%N A271363 Irregular triangle read by rows: T(0, 0) = 2; T(i, j) is the j-th term in the least maximal chain of composites that is longer than the (i-1)-st least maximal chain of composites, where i>0.
%C A271363 Call a sequence a_0, ..., a_k, k>=0, such that a_0 is even, a_k is prime, a_1,...,a_(k-1) are composite and a_(i+1) = 2*A065855(a_i) + 1 for 0 <= i < k a maximal chain of composites.
%C A271363 Since A065855 is nondecreasing the rows and columns of the triangle, respectively, are increasing starting with row 2.
%C A271363 T(n,0) is the least number starting a maximal chain of composites that is longer than the chain in row n-1.
%C A271363 T(n,j) = 2*A065855(T(n,j-1)) + 1 for n>=0, j>0 and T(n,j-1) composite.
%C A271363 Are there infinitely many rows? Are there rows of infinite length? (see A263570)
%H A271363 Hartmut F. W. Hoft, <a href="/A271363/b271363.txt">Table of n, a(n) for n = 0..88</a>
%e A271363 a(0) = T(0, 0) = 2 since 2 is an even prime.
%e A271363 a(5) = T(2,2) = 17 since 2*A065855(2*A065855(T(2,0))+1)+1 = 2*A065855(2*A065855(14)+1)+1 = 2*A065855(2*7+1)+1 = 2*A065855(15)+1 = 2*8+1 = 17 and the maximal chain of composites starting at 14 is the first of length 3.
%e A271363 The triangle T(i, j) with complete rows 0..6 and parts of rows 7 and 8:
%e A271363 --------------------------------------------------------------------------
%e A271363 i\j  0   1    2    3    4    5     6     7     8     9     10     11  ...
%e A271363 --------------------------------------------------------------------------
%e A271363 0:   2
%e A271363 1:   4   3
%e A271363 2:  14  15   17
%e A271363 3:  18  21   25   31
%e A271363 4:  40  55   77  111  163
%e A271363 5:  50  69   99  147  225  353
%e A271363 6:  60  85  123  185  285  447   721  1185  1981  3363   5777  10039
%e A271363 7:  82 119  177  273  429  693  1135  1891  3201  5497   9543  16723  ...
%e A271363 8: 490 793 1309 2189 3723 6407 11145 19591 34737 62055 111633 202093  ...
%e A271363 The entire right boundary of the triangle is A263570.
%e A271363 All numbers in the triangle through T(8, 31) can be found in the link.
%t A271363 (* a271363[n] computes a maximal chain of composites starting at n *)
%t A271363 composites[{m_, n_}] := Module[{i, count=0}, For[i=m, i<=n, i++, If[CompositeQ[i], count++]]; count]
%t A271363 a271363[n_] := Module[{i=n, j=composites[{0, n}], h, list={}}, While[CompositeQ[i], AppendTo[list, {i, j}]; h=composites[{i, 2*j+1}]; i=2*j+1; j+=h-1]; AppendTo[list, {i, j}]]
%t A271363 Map[First, ax271363[82]] (* computes row 7 *)
%Y A271363 Cf. A065855, A073846, A263570.
%K A271363 nonn,tabf
%O A271363 0,1
%A A271363 _Hartmut F. W. Hoft_, Apr 05 2016