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A253196
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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).
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1
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Dirichlet g.f. for column k: 1/k^(2*s) * zeta(s)^2/zeta(2*s).
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EXAMPLE
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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.
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MAPLE
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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))):
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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