A272210 - OEIS

%I #33 Jul 02 2016 00:49:33

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

%C This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th antidiagonal (read upwards) of the difference table 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 diagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of diagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].

%C First differs from A273132 at a(89).

%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 upwards gives the finite subsequence [1], [1, 2], [0, 1, 3], [2, 2, 3, 6], [-4, -2, 0, 3, 9], [12, 8, 6, 6, 9, 18].

%t Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* _Michael De Vlieger_, Jun 29 2016 *)

%Y Cf. A000005, A000217, A027750, A161700, A184389, A187202, A273102, A273103, A273109, A273135, A273132, A273136,  A273261, A273262, A273263.

%K sign,tabf

%O 1,4

%A _Omar E. Pol_, May 18 2016