A273135 - OEIS

%I #32 Jun 29 2016 00:02:46

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

%C 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 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 number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).

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

%C The antidiagonal sums give A273262.

%C 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 andiagonals is [1], [2, 1], [4, 2, 1], [8, 4, 2, 1].

%C First differs from A272121 at a(92).

%e The tables of the first nine positive integers are

%e 1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;

%e .  1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;

%e .              1;             0, 2;             1, 2;       4;

%e .                             2;                1;

%e .

%e For n = 18 the difference table 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 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 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 Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.

%K sign,tabf

%O 1,3

%A _Omar E. Pol_, May 18 2016