cp's OEIS Frontend

A273102 Difference table of the divisors of the positive integers.

%I A273102 #64 Feb 27 2020 09:20:50
%S A273102 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 A273102 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 A273102 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 A273102 Difference table of the divisors of the positive integers.
%C A273102 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 A273102 The first row of the slice n is also the n-th row of the triangle A027750.
%C A273102 The bottom entry of the slice n is A187202(n).
%C A273102 The sum of the elements of the slice n is A273103(n).
%C A273102 For another version see A273104, from which differs at a(92).
%C A273102 From _David A. Corneth_, May 20 2016: (Start)
%C A273102 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 A273102 a
%C A273102 . . b-a
%C A273102 b . . . . c-2b+a
%C A273102 . . c-b . . . . . d-3c+3b-a
%C A273102 c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
%C A273102 . . d-c . . . . . e-3d+3c-b
%C A273102 d . . . . e-2d+c
%C A273102 . . e-d
%C A273102 e
%C A273102 From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
%C A273102 (End)
%e A273102 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 A273102   1 . 2 . 3 . 6 . 9 . 18
%e A273102     1 . 1 . 3 . 3 . 9
%e A273102       0 . 2 . 0 . 6
%e A273102         2 .-2 . 6
%e A273102          -4 . 8
%e A273102            12
%e A273102 and the 18th slice is
%e A273102   1, 2, 3, 6, 9, 18;
%e A273102   1, 1, 3, 3, 9;
%e A273102   0, 2, 0, 6;
%e A273102   2,-2, 6;
%e A273102   -4, 8;
%e A273102   12;
%e A273102 The tetrahedron begins:
%e A273102   1;
%e A273102   1, 2;
%e A273102   1;
%e A273102   1, 3;
%e A273102   2;
%e A273102   1, 2, 4;
%e A273102   1, 2;
%e A273102   1;
%e A273102   ...
%e A273102 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 A273102 Triangle begins:
%e A273102   1;
%e A273102   1, 2, 1;
%e A273102   1, 3, 2;
%e A273102   1, 2, 4, 1, 2, 1;
%e A273102   ...
%t A273102 Table[Drop[FixedPointList[Differences, Divisors@ n], -2], {n, 15}] // Flatten (* _Michael De Vlieger_, May 16 2016 *)
%o A273102 (Sage)
%o A273102 def A273102_DTD(n): # DTD = Difference Table of Divisors
%o A273102     D = divisors(n)
%o A273102     T = matrix(ZZ, len(D))
%o A273102     for (m, d) in enumerate(D):
%o A273102         T[0, m] = d
%o A273102         for k in range(m-1, -1, -1) :
%o A273102             T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
%o A273102     return [T.row(k)[:len(D)-k] for k in range(len(D))]
%o A273102 # Keeps the rows of the DTD, for instance
%o A273102 # A273102_DTD(18)[1] = 1,1,3,3,9 (see the example above).
%o A273102 for n in range(1,19): print(A273102_DTD(n)) # _Peter Luschny_, May 18 2016
%Y A273102 Cf. A007182, A027750, A184389, A187202, A187204, A273103, A273104, A273109.
%K A273102 sign,tabf
%O A273102 1,3
%A A273102 _Omar E. Pol_, May 15 2016