cp's OEIS Frontend

A273136 Difference table of the divisors of the positive integers (with every table read by columns).

%I A273136 #14 Jun 29 2016 00:03:00
%S A273136 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 A273136 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 A273136 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 A273136 Difference table of the divisors of the positive integers (with every table read by columns).
%C A273136 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 A273136 The first row of the slice n is also the n-th row of the triangle A027750.
%C A273136 The bottom entry of the slice n is A187202(n).
%C A273136 The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
%C A273136 The sum of the elements of the n-th slice is A273103(n).
%C A273136 The columns sums give A273263.
%C A273136 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 A273136 First differs from A273137 at a(86).
%e A273136 The tables of the first nine positive integers are
%e A273136 1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
%e A273136 .  1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
%e A273136 .              1;             0, 2;             1, 2;       4;
%e A273136 .                             2;                1;
%e A273136 .
%e A273136 For n = 18 the difference table of the divisors of 18 is
%e A273136 1, 2, 3, 6, 9, 18;
%e A273136 1, 1, 3, 3, 9;
%e A273136 0, 2, 0, 6;
%e A273136 2, -2, 6;
%e A273136 -4, 8;
%e A273136 12;
%e A273136 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 A273136 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 A273136 Cf. A000005, A000217, A027750, A184389, A187202, A272210, A273102, A273103, A273135, A273137, A273263.
%K A273136 sign,tabf
%O A273136 1,4
%A A273136 _Omar E. Pol_, Jun 26 2016