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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.
2

%I #8 Apr 07 2016 02:58:37

%S 2,4,3,14,15,17,18,21,25,31,40,55,77,111,163,50,69,99,147,225,353,60,

%T 85,123,185,285,447,721,1185,1981,3363,5777,10039,82,119,177,273,429,

%U 693,1135,1891,3201,5497,9543,16723,29579,52737,94705,171147,311101

%N 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 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 Since A065855 is nondecreasing the rows and columns of the triangle, respectively, are increasing starting with row 2.

%C T(n,0) is the least number starting a maximal chain of composites that is longer than the chain in row n-1.

%C T(n,j) = 2*A065855(T(n,j-1)) + 1 for n>=0, j>0 and T(n,j-1) composite.

%C Are there infinitely many rows? Are there rows of infinite length? (see A263570)

%H Hartmut F. W. Hoft, <a href="/A271363/b271363.txt">Table of n, a(n) for n = 0..88</a>

%e a(0) = T(0, 0) = 2 since 2 is an even prime.

%e 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 The triangle T(i, j) with complete rows 0..6 and parts of rows 7 and 8:

%e --------------------------------------------------------------------------

%e i\j 0 1 2 3 4 5 6 7 8 9 10 11 ...

%e --------------------------------------------------------------------------

%e 0: 2

%e 1: 4 3

%e 2: 14 15 17

%e 3: 18 21 25 31

%e 4: 40 55 77 111 163

%e 5: 50 69 99 147 225 353

%e 6: 60 85 123 185 285 447 721 1185 1981 3363 5777 10039

%e 7: 82 119 177 273 429 693 1135 1891 3201 5497 9543 16723 ...

%e 8: 490 793 1309 2189 3723 6407 11145 19591 34737 62055 111633 202093 ...

%e The entire right boundary of the triangle is A263570.

%e All numbers in the triangle through T(8, 31) can be found in the link.

%t (* a271363[n] computes a maximal chain of composites starting at n *)

%t composites[{m_, n_}] := Module[{i, count=0}, For[i=m, i<=n, i++, If[CompositeQ[i], count++]]; count]

%t 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 Map[First, ax271363[82]] (* computes row 7 *)

%Y Cf. A065855, A073846, A263570.

%K nonn,tabf

%O 0,1

%A _Hartmut F. W. Hoft_, Apr 05 2016