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A357698
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a(n) is the sum of the aliquot divisors of n that are cubefree.
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2
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0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 7, 1, 21, 1, 22, 11, 14, 1, 28, 6, 16, 13, 28, 1, 42, 1, 7, 15, 20, 13, 55, 1, 22, 17, 42, 1, 54, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 39, 17, 56, 23, 32, 1, 108, 1, 34, 41, 7, 19, 78, 1, 58, 27, 74, 1, 91, 1, 40
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{d|n, d<n} A212793(d)*d.
a(n) = 1 iff n is a prime.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .
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EXAMPLE
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The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.
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MATHEMATICA
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f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;; , 2]], # < 3 &], n, 0]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i, 1] + if(f[i, 2] == 1, 0, f[i, 1]^2)); if(n==1 || vecmax(f[, 2]) < 3, s -= n); s};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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