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).

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, 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, 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

Offset: 1

Author

Geoffrey Critzer, Mar 24 2015

Keywords

• nonn
• tabf

Comments

Row sums are A000005.

Column 1 is A034444.

Examples

			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
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.
		

Links

• Alois P. Heinz, Rows n = 1..6000, flattened

Crossrefs

Cf. A000005, A000188, A034444.

Programs

Maple

with(numtheory):
T:= n-> (p-> seq(coeff(p, x, j), j=1..degree(p)))(add(
    x^mul(i[1]^iquo(i[2], 2), i=ifactors(d)[2]), d=divisors(n))):
seq(T(n), n=1..70);  # Alois P. Heinz, Mar 25 2015

Mathematica

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

Formula

Dirichlet g.f. for column k: 1/k^(2*s) * zeta(s)^2/zeta(2*s).