A273137 -id:A273137 - cp's OEIS frontend

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

Omar E. Pol, Jun 26 2016

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).

			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].

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

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

A274533 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th column of the absolute difference table of the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 4, 5, 6, 6, 7, 7, 4, 6, 8, 8, 7, 9, 9, 4, 7, 10, 10, 11, 11, 4, 6, 8, 10, 12, 12, 13, 13, 8, 9, 14, 14, 11, 13, 15, 15, 5, 8, 12, 16, 16, 17, 17, 8, 11, 12, 15, 18, 18, 19, 19, 7, 10, 10, 15, 20, 20, 13, 17, 21, 21, 16, 13, 22, 22, 23, 23, 6, 7, 10, 12, 16, 20, 24, 24, 21, 25, 25

Offset: 1

Omar E. Pol, Jun 29 2016

If n is prime then row n is [n, n].
It appears that the last two terms of the n-th row are [n, n], n > 1.
Note that this sequence is not the absolute values of A273263.
First differs from A273263 at a(38).

			Triangle begins:
   1;
   2,  2;
   3,  3;
   3,  4,  4;
   5,  5;
   4,  5,  6,  6;
   7,  7;
   4,  6,  8,  8;
   7,  9,  9;
   4,  7, 10, 10;
  11, 11;
   4,  6,  8, 10, 12, 12;
  13, 13;
   8,  9, 14, 14;
  11, 13, 15, 15;
   5,  8, 12, 16, 16;
  17, 17;
   8, 11, 12, 15, 18, 18;
  19, 19;
   7, 10, 10, 15, 20, 20;
  13, 17, 21, 21;
  16, 13, 22, 22;
  23, 23;
   6,  7, 10, 12, 16, 20, 24, 24;
  21, 25, 25;
  20, 15, 26, 26;
  ...
For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the absolute difference triangle of the divisors is
  1,  2,  3,  6,  9, 18;
  1,  1,  3,  3,  9;
  0,  2,  0,  6;
  2,  2,  6;
  0,  4;
  4;
The column sums give [8, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Right border gives A000027. Row sums give A187215.

Cf. A187203, A272121, A273132, A273104, A273137, A273263, A274531, A274532.

Table[Total /@ Table[#[[m - k + 1, -k]], {m, Length@ #, 1, -1}, {k, m}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 25}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica