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%I #26 Aug 27 2023 04:23:00
%S 1,3,4,7,6,12,8,14,13,18,12,28,14,24,24,28,18,39,20,42,32,36,24,56,31,
%T 42,39,56,30,72,32,56,48,54,48,91,38,60,56,84,42,96,44,84,78,72,48,
%U 112,57,93,72,98,54,117,72,112,80,90,60,168,62,96,104,112,84,144
%N a(n) is the sum of the divisors d of n such that n/d is cubefree.
%C Inverse Möbius transform of A254926.
%H <a href="/A254981/b254981.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d | n} d * A212793(n/d) = n * Sum_{d | n} A212793(d) / d.
%F a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3).
%F Multiplicative with a(p) = 1 + p; a(p^e) = p^(e-2) * (1 + p + p^2), for e>1.
%F Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(3s).
%F If n is powerful, a(n^k) = n^(k-1) * a(n).
%F For k>1, a(n^k) = n^(k-1) * a(n) * Product_{p prime, ord(n,p)=1} (p^3-1) / (p^3-p).
%F Sum_{k=1..n} a(k) ~ 315*n^2 / (4*Pi^4). - _Vaclav Kotesovec_, Feb 03 2019
%t nn = 66; f[list_, i_] := list[[i]]; a = Table[If[Max[FactorInteger[n][[All, 2]]] < 3, 1, 0], {n, 1, nn}]; b =Table[n, {n, 1, nn}]; Table[
%t DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* _Geoffrey Critzer_, Feb 22 2015 *)
%t f[p_, e_] := p^(e-2) * (1 + p + p^2); f[p_, 1] := 1 + p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)
%o (PARI) a212793(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); }
%o a(n) = sumdiv(n, d, d*a212793(n/d)); \\ _Michel Marcus_, Feb 11 2015
%o (PARI) a(n) = sumdiv(n, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n/d), 0)); \\ _Michel Marcus_, Mar 04 2015
%Y Cf. A000203, A001615, A001694, A212793, A254926.
%K mult,nonn,easy
%O 1,2
%A _Álvar Ibeas_, Feb 11 2015
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