cp's OEIS Frontend

A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

%I A273135 #36 May 01 2025 08:29:17
%S A273135 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,
%T A273135 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,
%U A273135 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
%N A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).
%C A273135 This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the difference triangle of the divisors of n.
%C A273135 The first row of the slice n is also the n-th row of the triangle A027750.
%C A273135 The bottom entry of the slice n is A187202(n).
%C A273135 The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
%C A273135 The sum of the elements of the n-th slice is A273103(n).
%C A273135 The antidiagonal sums give A273262.
%C A273135 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].
%C A273135 First differs from A272121 at a(92).
%e A273135 The tables of the first nine positive integers are
%e A273135   1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
%e A273135      1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
%e A273135                  1;             0, 2;             1, 2;       4;
%e A273135                                 2;                1;
%e A273135 For n = 18 the difference table of the divisors of 18 is
%e A273135   1,  2, 3, 6, 9, 18;
%e A273135   1,  1, 3, 3, 9;
%e A273135   0,  2, 0, 6;
%e A273135   2, -2, 6;
%e A273135  -4,  8;
%e A273135  12;
%e A273135 This table read by antidiagonals downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, -2, -4], [18, 9, 6, 6, 8, 12].
%t A273135 Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m, 1, -1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* _Michael De Vlieger_, Jun 26 2016 *)
%Y A273135 Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.
%K A273135 sign,tabf
%O A273135 1,3
%A A273135 _Omar E. Pol_, May 18 2016