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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). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums are A000005.

Column 1 is A034444.

LINKS

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

FORMULA

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

EXAMPLE

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.

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

CROSSREFS

Cf. A000005, A000188, A034444.

Sequence in context: A029330 A132225 A263923 * A271205 A303841 A093116

Adjacent sequences:  A253193 A253194 A253195 * A253197 A253198 A253199

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Mar 24 2015

STATUS

approved

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Last modified October 22 10:50 EDT 2018. Contains 316436 sequences. (Running on oeis4.)