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

1,2

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

Links

Table of n, a(n) for n=1..86.

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

Formula

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

Example

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,...

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 *)

Crossrefs

Columns (1,2): A000005, A183063.

Cf. A051731, A127170.

Sequence in context: A096810 A190436 A133696 * A127371 A036849 A097364

Adjacent sequences: A195047 A195048 A195049 * A195051 A195052 A195053

Keyword

nonn,tabl

Author

Omar E. Pol, Oct 18 2011

Status

approved