A273137 Absolute 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

1,4

Comments

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column of the absolute difference table 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 subsequence lists the elements of the absolute 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].

Note that this sequence is not the absolute values of A273136.

First differs from A273136 at a(86).

Links

Table of n, a(n) for n=1..102.

Example

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 columns gives the finite subsequence [1, 1, 0, 2, 0, 4], [2, 1, 2, 2, 4], [3, 3, 0, 6], [6, 3, 6], [9, 9], [18].

Mathematica

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

Crossrefs

Cf. A000005, A000217, A027750, A184389, A187203, A187215, A272121, A273132, A273136.

Sequence in context: A143343 A138243 A273136 * A362947 A246064 A238157

Adjacent sequences: A273134 A273135 A273136 * A273138 A273139 A273140

Keyword

nonn,tabf

Author

Omar E. Pol, Jun 26 2016

Status

approved