A073185 Sum of cubefree divisors of n.

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68, 126, 96, 144, 72

Offset

1,2

Comments

Sum of divisors of the cubefree kernel of n (see first formula).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000203(A007948(n)).

a(n) <= A073183(n).

Multiplicative with a(p) = 1+p, a(p^e) = 1 + p + p^2, e>1. - Christian G. Bower, May 18 2005

a(n) = sum(A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)). - Reinhard Zumkeller, May 27 2012

Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(3s-3). - R. J. Mathar, Apr 12 2011

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*Zeta(3)). - Vaclav Kotesovec, Feb 01 2019

Example

The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, 8=2^3 and 56=7*2^3 are not cubefree, therefore a(56) = 1 + 2 + 4 + 7 + 14 + 28 = 56.

Maple

charFfree := proc(n, t) local f; for f in ifactors(n)[2] do if op(2, f) >= t then return 0 ; end if; end do: return 1 ; end proc:
A073185 := proc(n) add( d*charFfree(d, 3), d =numtheory[divisors](n) ); end proc: # R. J. Mathar, Apr 12 2011

Mathematica

nn = 71; f[list_, i_] := list[[i]]; a =Table[If[Max[FactorInteger[n][[All, 2]]] <= 2, n, 0], {n, 1, nn}]; b = Table[1, {nn}]; Select[Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}], # > 0 &] (* Geoffrey Critzer, Mar 22 2015 *)
f[p_, e_] := 1 + p + If[e > 1, p^2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

Prog

(Haskell)
a073185 = sum . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012
(PARI) a(n) = {my(f=factor(n)); for (i=1, #f~, p = f[i, 1]; if ((e=f[i, 2]) == 1, f[i, 1] = 1+p, f[i, 1] = 1+p+p^2); f[i, 2] = 1; ); factorback(f); } \\ Michel Marcus, Feb 06 2015

Crossrefs

Cf. A000203, A073184, A004709, A073182, A073181, A048250.

Sequence in context: A367171 A348946 A366744 * A284341 A073183 A353900

Adjacent sequences: A073182 A073183 A073184 * A073186 A073187 A073188

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jul 19 2002

Extensions

Incorrect comment removed by Álvar Ibeas, Feb 06 2015

Status

approved