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
Examples
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.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Programs
-
Haskell
a073185 = sum . filter ((== 1) . a212793) . a027750_row -- Reinhard Zumkeller, May 27 2012
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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
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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 *) -
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
Formula
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
Extensions
Incorrect comment removed by Álvar Ibeas, Feb 06 2015
Comments