A273136 Difference table of the divisors of the positive integers (with every table read by columns).

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

Offset: 1

Author

Omar E. Pol, Jun 26 2016

Keywords

• sign
• tabf

Comments

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column of the 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 A187202(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 A273103(n).

The columns sums give A273263.

If n is a power of 2 the subsequence lists the elements of the difference table of the divisors of n in nondecreasing order, for example if n = 8 the finite sequence of columns is [1, 1, 1, 1], [2, 2, 2], [4, 4], [8].

First differs from A273137 at a(86).

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 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;
-4, 8;
12;
This table read by columns gives the finite subsequence [1, 1, 0, 2, -4, 12], [2, 1, 2, -2, 8], [3, 3, 0, 6], [6, 3, 6], [9, 9], [18].
		

Crossrefs

Cf. A000005, A000217, A027750, A184389, A187202, A272210, A273102, A273103, A273135, A273137, A273263.

Programs

Mathematica

Table[Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 15}] /. 0 -> {} // Flatten (* Michael De Vlieger, Jun 26 2016 *)