A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

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

Offset

1,3

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 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 antidiagonal sums give A273262.

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

First differs from A272121 at a(92).

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 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 antidiagonals downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, -2, -4], [18, 9, 6, 6, 8, 12].

Mathematica

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

Crossrefs

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.

Sequence in context: A355737 A112380 A272121 * A165162 A125106 A229874

Adjacent sequences: A273132 A273133 A273134 * A273136 A273137 A273138

Keyword

sign,tabf

Author

Omar E. Pol, May 18 2016

Status

approved