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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, listed by rows.

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

----------------------------------------------------------------------------

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i\j 0 1 2 3 4 5 6 ... 7 8 9 10 11 12 ... 23 ... 31

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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 29579 . 23023727..

8: 490 793 1309 2189 3723 6407 11145 . 19591 34737 62055 111633 202093 367873 .. 344139423 . 59331187155

The entire right boundary of the triangle is A263570.

nonn,changed,tabf

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

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 . 10039

82 119 177 273 429 693 1135 . 16723 29579 . 23023727

(* 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 *)

allocated for Hartmut F. W. Hoft

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, i>0, listed by rows.

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

0,1

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)

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...7 and parts of row 8:

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0 1 2 3 4 5 6 ... 11 12 ... 23 ... 31

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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 . 10039

82 119 177 273 429 693 1135 . 16723 29579 . 23023727

490 793 1309 2189 3723 6407 11145 . 202093 367873 . 344139423 . 59331187155

The right boundary of the triangle is A263570.

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

Cf. A065855, A073846, A263570.

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Hartmut F. W. Hoft, Apr 05 2016

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allocated for Hartmut F. W. Hoft

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