A273104 -id:A273104 - cp's OEIS frontend

A273102 Difference table of the divisors of the positive integers.

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

Offset: 1

Omar E. Pol, May 15 2016

This is an irregular tetrahedron T(n,j,k) read by rows in which the slice n lists the elements of the rows of the difference triangle of the divisors of n (including 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 sum of the elements of the slice n is A273103(n).
For another version see A273104, from which differs at a(92).
From David A. Corneth, May 20 2016: (Start)
Each element of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the difference triangle 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
and the 18th slice is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2,-2, 6;
  -4, 8;
  12;
The tetrahedron begins:
  1;
  1, 2;
  1;
  1, 3;
  2;
  1, 2, 4;
  1, 2;
  1;
  ...
This is also an irregular triangle T(n,r) read by rows in which row n lists the difference triangle of the divisors of n flattened. Row lengths are the terms of A184389. Row sums give A273103.
Triangle begins:
  1;
  1, 2, 1;
  1, 3, 2;
  1, 2, 4, 1, 2, 1;
  ...

Cf. A007182, A027750, A184389, A187202, A187204, A273103, A273104, A273109.

Table[Drop[FixedPointList[Differences, Divisors@ n], -2], {n, 15}] // Flatten (* Michael De Vlieger, May 16 2016 *)

def A273102_DTD(n): # DTD = Difference Table of Divisors
    D = divisors(n)
    T = matrix(ZZ, len(D))
    for (m, d) in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return [T.row(k)[:len(D)-k] for k in range(len(D))]
# Keeps the rows of the DTD, for instance
# A273102_DTD(18)[1] = 1,1,3,3,9 (see the example above).
for n in range(1,19): print(A273102_DTD(n)) # Peter Luschny, May 18 2016

A272121 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

Original entry on oeis.org

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

Omar E. Pol, May 18 2016

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 in decreasing order, for example if n = 8 the finite sequence of antidiagonals is [1], [2, 1], [4, 2, 1], [8, 4, 2, 1].
First differs from A273135 at a(92).
Note that this sequence is not the absolute values of A273135. For example a(135) = 0 and A273135(135) = 4.

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

Cf. A000005, A000217, A027750, A184389, A187203, A187215, A273104, A273132, A273135.

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

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

Original entry on oeis.org

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

Omar E. Pol, May 18 2016

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.

			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].

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

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 *)

A274531 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the absolute difference table of the divisors of n.

Original entry on oeis.org

1, 3, 1, 4, 2, 7, 3, 1, 6, 4, 12, 5, 2, 2, 8, 6, 15, 7, 3, 1, 13, 8, 4, 18, 9, 4, 0, 12, 10, 28, 11, 5, 4, 3, 1, 14, 12, 24, 13, 6, 2, 24, 14, 8, 8, 31, 15, 7, 3, 1, 18, 16, 39, 17, 8, 10, 4, 4, 20, 18, 42, 19, 11, 4, 5, 1, 32, 20, 12, 8, 36, 21, 10, 6, 24, 22, 60, 23, 11, 10, 6, 5, 2, 2, 31, 24, 16, 42, 25, 12, 8

Offset: 1

Omar E. Pol, Jun 27 2016

Row 2^k gives the first k+1 positive terms of A000225 in decreasing order, k >= 0.
If n is prime then row n contains only two terms: n+1 and n-1.
Note that this sequence is not the absolute values of A273261.
First differs from A273261 at a(41).

			Triangle begins:
1;
3, 1;
4, 2;
7, 3, 1;
6, 4;
12, 5, 2, 2;
8, 6;
15, 7, 3, 1;
13, 8, 4;
18, 9, 4, 0;
12, 10;
28, 11, 5, 4, 3, 1;
14, 12;
24, 13, 6, 2;
24, 14, 8, 8;
31, 15, 7, 3, 1;
18, 16;
39, 17, 8, 10, 4, 4;
20, 18;
42, 19, 11, 4, 5, 1;
32, 20, 12, 8;
36, 21, 10, 6;
24, 22;
60, 23, 11, 10, 6, 5, 2, 2;
31, 24, 16;
42, 25, 12, 8;
...
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 row sums give [39, 17, 8, 10, 4, 4] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Column 1 is A000203. Right border gives A187203. Row sums give A187215.

Cf. A000225, A273104, A273261, A274532, A274533.

Map[Total, #, {2}] &@ Table[NestWhileList[Abs@ Differences@ # &, #, Length@ # > 1 &] &@ Divisors@ n, {n, 26}] // Flatten (* Michael De Vlieger, Jun 27 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

A273102 Difference table of the divisors of the positive integers.

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A272121 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

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

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A274531 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the absolute difference table of the divisors of n.

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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.

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