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A253196 Irregular array read by rows. T(n,k) is the number of divisors d of n such that k^2 is the greatest square that divides d, n>=1, 1<=k<=A000188(n).

%I A253196 #20 Mar 25 2015 18:43:09
%S A253196 1,2,2,2,1,2,4,2,2,2,2,0,1,4,2,4,2,2,4,4,2,2,0,1,2,4,0,2,2,4,2,4,4,2,
%T A253196 4,4,2,0,0,0,1,4,2,0,2,4,2,2,8,2,2,2,0,2,4,4,4,4,2,2,0,0,1,2,4,4,4,4,
%U A253196 2,8,2,4,2,4,0,2,4,2,4,4,0,2,2,0,0,0,0,0,1,4,0,0,0,2,4,4,2,2,4,0,4,4,4,4,4,4,2,8,4
%N A253196 Irregular array read by rows.  T(n,k) is the number of divisors d of n such that k^2 is the greatest square that divides d, n>=1, 1<=k<=A000188(n).
%C A253196 Row sums are A000005.
%C A253196 Column 1 is A034444.
%H A253196 Alois P. Heinz, <a href="/A253196/b253196.txt">Rows n = 1..6000, flattened</a>
%F A253196 Dirichlet g.f. for column k: 1/k^(2*s) * zeta(s)^2/zeta(2*s).
%e A253196 1
%e A253196 2
%e A253196 2
%e A253196 2,1
%e A253196 2
%e A253196 4
%e A253196 2
%e A253196 2,2
%e A253196 2,0,1
%e A253196 4
%e A253196 2
%e A253196 4,2
%e A253196 2
%e A253196 4
%e A253196 4
%e A253196 2,2,0,1
%e A253196 2
%e A253196 4,0,2
%e A253196 For n=18, The divisors are: 1,2,3,6,9,18.  T(18,1)=4 because 1 is the largest square that divides 1,2,3,6.  T(18,3) = 2 because 9 is the largest square that divides 9,18.
%p A253196 with(numtheory):
%p A253196 T:= n-> (p-> seq(coeff(p, x, j), j=1..degree(p)))(add(
%p A253196     x^mul(i[1]^iquo(i[2], 2), i=ifactors(d)[2]), d=divisors(n))):
%p A253196 seq(T(n), n=1..70);  # _Alois P. Heinz_, Mar 25 2015
%t A253196 nn = 60;g[list_] := list /. {j___, 0 ...} -> {j}; f[list_, i_] := list[[i]];Map[g, Transpose[Table[a = Table[If[n == k^2, 1, 0], {n, 1, nn}]; b = Table[2^PrimeNu[n], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}], {k,1, nn}]]] // Grid
%Y A253196 Cf. A000005, A000188, A034444.
%K A253196 nonn,tabf
%O A253196 1,2
%A A253196 _Geoffrey Critzer_, Mar 24 2015