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A073185 Sum of cubefree divisors of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A279388 A292288 A113957 * A284341 A073183 A049418

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

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Last modified April 16 21:58 EDT 2021. Contains 343051 sequences. (Running on oeis4.)