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A273137
Absolute difference table of the divisors of the positive integers (with every table read by columns).
2
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, 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, 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
OFFSET
1,4
COMMENTS
This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column of the absolute difference table of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187203(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A187215(n).
If n is a power of 2 the subsequence lists the elements of the absolute 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].
Note that this sequence is not the absolute values of A273136.
First differs from A273136 at a(86).
EXAMPLE
The tables of the first nine positive integers are
1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
. 1; 2; 1, 2; 4; 1, 1, 3; 6; 1, 2, 4; 2, 6;
. 1; 0, 2; 1, 2; 4;
. 2; 1;
.
For n = 18 the absolute difference table of the divisors of 18 is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, 2, 6;
0, 4;
4;
This table read by columns gives the finite subsequence [1, 1, 0, 2, 0, 4], [2, 1, 2, 2, 4], [3, 3, 0, 6], [6, 3, 6], [9, 9], [18].
MATHEMATICA
Table[Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Abs@ Differences@ # &, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 15}] /. 0 -> {} // Flatten (* Michael De Vlieger, Jun 26 2016 *)
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jun 26 2016
STATUS
approved