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

Author

Omar E. Pol, May 15 2016

Keywords

• sign
• tabf

Comments

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)

Examples

			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;
  ...
		

Crossrefs

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

Programs

Mathematica

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

Sage

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