A195050 Square array T(n,k) read by antidiagonals in which column k lists the number of divisors of n that are divisible by k.

1, 2, 0, 2, 1, 0, 3, 0, 0, 0, 2, 2, 1, 0, 0, 4, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 1, 0

Offset: 1

Author

Omar E. Pol, Oct 18 2011

Keywords

• nonn
• tabl

Comments

It appears that the sequence formed by starting with an initial set of k-1 zeros followed by the members of A000005, with k-1 zeros between every one of them, can be defined as "the number of divisors of n that are divisible by k", (k >= 1). For example: if k = 1 we have A000005 by definition; if k = 2 we have A183063. Note that if k >= 3 the sequences are not included in the OEIS because the usual OEIS policy is not to include sequences with interspersed zeros. A183063 is an exception.

It appears that the illustration of initial terms of column k can be represented by a general diagram in which the period of the smallest curve is 2*k, hence the distance between consecutive two nodes is equal to k. (For k = 1 see the link.)

Row sums = A007425. - Geoffrey Critzer, Feb 07 2015

Examples

			Array begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,...
2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,...
4, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0,...
2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,...
4, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0,...
3, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0,...
4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0,...
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,...
6, 4, 3, 2, 0, 2, 0, 0, 0, 0, 0, 1,...
		

Links

• Omar E. Pol, Illustration of initial terms of column 1

Crossrefs

Columns (1,2): A000005, A183063.

Cf. A051731, A127170.

Programs

Mathematica

(* returns square array *)
nn = 20; Transpose[Table[Table[DirichletConvolve[1, Floor[n/k] - Floor[(n - 1)/k], n, m], {m, 1,nn}], {k, 1, nn}]] // Grid (* Geoffrey Critzer, Feb 07 2015 *)

Formula

Dirichlet generating function of column k: zeta(s)*Sum_{n>=1}1/(k*n)^s. - Geoffrey Critzer, Feb 07 2015