A273102 - OEIS

%I #64 Feb 27 2020 09:20:50

%S 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,

%T 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,

%U 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

%N Difference table of the divisors of the positive integers.

%C 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).

%C The first row of the slice n is also the n-th row of the triangle A027750.

%C The bottom entry of the slice n is A187202(n).

%C The sum of the elements of the slice n is A273103(n).

%C For another version see A273104, from which differs at a(92).

%C From _David A. Corneth_, May 20 2016: (Start)

%C 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):

%C a

%C . . b-a

%C b . . . . c-2b+a

%C . . c-b . . . . . d-3c+3b-a

%C c . . . . d-2c+b . . . . . . e-4d+6c-4b+a

%C . . d-c . . . . . e-3d+3c-b

%C d . . . . e-2d+c

%C . . e-d

%C e

%C From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.

%C (End)

%e For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the difference triangle of the divisors of 18 is

%e   1 . 2 . 3 . 6 . 9 . 18

%e     1 . 1 . 3 . 3 . 9

%e       0 . 2 . 0 . 6

%e         2 .-2 . 6

%e          -4 . 8

%e            12

%e and the 18th slice is

%e   1, 2, 3, 6, 9, 18;

%e   1, 1, 3, 3, 9;

%e   0, 2, 0, 6;

%e   2,-2, 6;

%e   -4, 8;

%e   12;

%e The tetrahedron begins:

%e   1;

%e   1, 2;

%e   1;

%e   1, 3;

%e   2;

%e   1, 2, 4;

%e   1, 2;

%e   1;

%e   ...

%e 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.

%e Triangle begins:

%e   1;

%e   1, 2, 1;

%e   1, 3, 2;

%e   1, 2, 4, 1, 2, 1;

%e   ...

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

%o (Sage)

%o def A273102_DTD(n): # DTD = Difference Table of Divisors

%o     D = divisors(n)

%o     T = matrix(ZZ, len(D))

%o     for (m, d) in enumerate(D):

%o         T[0, m] = d

%o         for k in range(m-1, -1, -1) :

%o             T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]

%o     return [T.row(k)[:len(D)-k] for k in range(len(D))]

%o # Keeps the rows of the DTD, for instance

%o # A273102_DTD(18)[1] = 1,1,3,3,9 (see the example above).

%o for n in range(1,19): print(A273102_DTD(n)) # _Peter Luschny_, May 18 2016

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

%K sign,tabf

%O 1,3

%A _Omar E. Pol_, May 15 2016