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Difference table of the divisors of the positive integers (with every table read by columns).
3

%I #14 Jun 29 2016 00:03:00

%S 1,1,1,2,1,2,3,1,1,1,2,2,4,1,4,5,1,1,0,2,2,1,2,3,3,6,1,6,7,1,1,1,1,2,

%T 2,2,4,4,8,1,2,4,3,6,9,1,1,2,0,2,3,2,5,5,10,1,10,11,1,1,0,0,1,1,2,1,0,

%U 1,2,3,1,1,3,4,2,4,6,6,12,1,12,13,1,1,4,-2,2,5,2,7,7,14,1,2,0,8,3,2,8,5,10,15

%N Difference table of the divisors of the positive integers (with every table read by columns).

%C This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column 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 columns sums give A273263.

%C If n is a power of 2 the subsequence lists the elements of the difference table of the divisors of n in nondecreasing order, for example if n = 8 the finite sequence of columns is [1, 1, 1, 1], [2, 2, 2], [4, 4], [8].

%C First differs from A273137 at a(86).

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

%t Table[Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 15}] /. 0 -> {} // Flatten (* _Michael De Vlieger_, Jun 26 2016 *)

%Y Cf. A000005, A000217, A027750, A184389, A187202, A272210, A273102, A273103, A273135, A273137, A273263.

%K sign,tabf

%O 1,4

%A _Omar E. Pol_, Jun 26 2016