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.

2, 4, 3, 14, 15, 17, 18, 21, 25, 31, 40, 55, 77, 111, 163, 50, 69, 99, 147, 225, 353, 60, 85, 123, 185, 285, 447, 721, 1185, 1981, 3363, 5777, 10039, 82, 119, 177, 273, 429, 693, 1135, 1891, 3201, 5497, 9543, 16723, 29579, 52737, 94705, 171147, 311101

Offset: 0

Author

Hartmut F. W. Hoft, Apr 05 2016

Keywords

• nonn
• tabf

Comments

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.

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

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

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

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

Examples

			a(0) = T(0, 0) = 2 since 2 is an even prime.
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.
The triangle T(i, j) with complete rows 0..6 and parts of rows 7 and 8:
--------------------------------------------------------------------------
i\j  0   1    2    3    4    5     6     7     8     9     10     11  ...
--------------------------------------------------------------------------
0:   2
1:   4   3
2:  14  15   17
3:  18  21   25   31
4:  40  55   77  111  163
5:  50  69   99  147  225  353
6:  60  85  123  185  285  447   721  1185  1981  3363   5777  10039
7:  82 119  177  273  429  693  1135  1891  3201  5497   9543  16723  ...
8: 490 793 1309 2189 3723 6407 11145 19591 34737 62055 111633 202093  ...
The entire right boundary of the triangle is A263570.
All numbers in the triangle through T(8, 31) can be found in the link.
		

Links

• Hartmut F. W. Hoft, Table of n, a(n) for n = 0..88

Crossrefs

Cf. A065855, A073846, A263570.

Programs

Mathematica

(* a271363[n] computes a maximal chain of composites starting at n *)
composites[{m_, n_}] := Module[{i, count=0}, For[i=m, i<=n, i++, If[CompositeQ[i], count++]]; count]
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}]]
Map[First, ax271363[82]] (* computes row 7 *)