|
%I #42 Feb 11 2015 07:35:01
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,1,0
%N Square array T(n,k) read by antidiagonals in which column k lists the number of divisors of n that are divisible by k.
%C 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.
%C 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.)
%C Row sums = A007425. - _Geoffrey Critzer_, Feb 07 2015
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Illustration of initial terms of column 1</a>
%F Dirichlet generating function of column k: zeta(s)*Sum_{n>=1}1/(k*n)^s. - _Geoffrey Critzer_, Feb 07 2015
%e Array begins:
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%e 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%e 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%e 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,...
%e 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,...
%e 4, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0,...
%e 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,...
%e 4, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0,...
%e 3, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0,...
%e 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0,...
%e 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,...
%e 6, 4, 3, 2, 0, 2, 0, 0, 0, 0, 0, 1,...
%t (* returns square array *)
%t 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 *)
%Y Columns (1,2): A000005, A183063.
%Y Cf. A051731, A127170.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Oct 18 2011
|
