cp's OEIS Frontend

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

%I A273132 #19 May 01 2025 08:29:22
%S A273132 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,
%T A273132 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,
%U A273132 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
%N A273132 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).
%C A273132 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.
%C A273132 The first row of the slice n is also the n-th row of the triangle A027750.
%C A273132 The bottom entry of the slice n is A187203(n).
%C A273132 The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
%C A273132 The sum of the elements of the n-th slice is A187215(n).
%C A273132 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].
%C A273132 First differs from A272210 at a(89).
%C A273132 Note that this sequence is not the absolute values of A272210. For example a(131) = 0 and A272210(131) = 4.
%e A273132 The tables of the first nine positive integers are
%e A273132   1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
%e A273132      1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
%e A273132                  1;             0, 2;             1, 2;       4;
%e A273132                                 2;                1;
%e A273132 For n = 18 the absolute difference table of the divisors of 18 is
%e A273132   1, 2, 3, 6, 9, 18;
%e A273132   1, 1, 3, 3, 9;
%e A273132   0, 2, 0, 6;
%e A273132   2, 2, 6;
%e A273132   0, 4;
%e A273132   4;
%e A273132 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].
%t A273132 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 *)
%Y A273132 Cf. A000005, A000217, A027750, A184389, A187203, A187215, A272121, A273104.
%K A273132 nonn,tabf
%O A273132 1,4
%A A273132 _Omar E. Pol_, May 18 2016