A273132 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

1, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 4, 1, 4, 5, 1, 1, 2, 0, 1, 3, 2, 2, 3, 6, 1, 6, 7, 1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 3, 4, 6, 9, 1, 1, 2, 2, 3, 5, 0, 2, 5, 10, 1, 10, 11, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 1, 1, 1, 2, 6, 1, 2, 3, 4, 6, 12, 1, 12, 13, 1, 1, 2, 4, 5, 7, 2, 2, 7, 14, 1, 2, 3, 0, 2, 5, 8, 8, 10, 15

Offset: 1

Author

Omar E. Pol, May 18 2016

Keywords

• nonn
• tabf

Comments

This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the absolute difference triangle of the divisors of n.

The first row of the slice n is also the n-th row of the triangle A027750.

The bottom entry of the slice n is A187203(n).

The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).

The sum of the elements of the n-th slice is A187215(n).

If n is a power of 2 the antidiagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of antidiagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].

First differs from A272210 at a(89).

Note that this sequence is not the absolute values of A272210. For example a(131) = 0 and A272210(131) = 4.

Examples

			The tables of the first nine positive integers are
  1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
     1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
                 1;             0, 2;             1, 2;       4;
                                2;                1;
For n = 18 the absolute difference table of the divisors of 18 is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2, 2, 6;
  0, 4;
  4;
This table read by antidiagonals upwards gives the finite subsequence [1], [1, 2], [0, 1, 3], [2, 2, 3, 6], [0, 2, 0, 3, 9], [4, 4, 6, 6, 9, 18].
		

Crossrefs

Cf. A000005, A000217, A027750, A184389, A187203, A187215, A272121, A273104.

Programs

Mathematica

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)