A254981 a(n) is the sum of the divisors d of n such that n/d is cubefree.

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, 42, 39, 56, 30, 72, 32, 56, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 112, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 112, 84, 144

Offset: 1

Author

Álvar Ibeas, Feb 11 2015

Keywords

• mult
• nonn
• easy

Comments

Inverse Möbius transform of A254926.

Links

• Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A001615, A001694, A212793, A254926.

Programs

Mathematica

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[
DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)
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 *)

PARI

a212793(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); }
a(n) = sumdiv(n, d, d*a212793(n/d)); \\ Michel Marcus, Feb 11 2015

PARI

a(n) = sumdiv(n, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n/d), 0)); \\ Michel Marcus, Mar 04 2015

Formula

a(n) = Sum_{d | n} d * A212793(n/d) = n * Sum_{d | n} A212793(d) / d.

a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3).

Multiplicative with a(p) = 1 + p; a(p^e) = p^(e-2) * (1 + p + p^2), for e>1.

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(3s).

If n is powerful, a(n^k) = n^(k-1) * a(n).

For k>1, a(n^k) = n^(k-1) * a(n) * Product_{p prime, ord(n,p)=1} (p^3-1) / (p^3-p).

Sum_{k=1..n} a(k) ~ 315*n^2 / (4*Pi^4). - Vaclav Kotesovec, Feb 03 2019