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 A369306 The number of cubefree divisors d of n such that n/d is also cubefree. 1
 1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 0, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 0, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)). LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5. a(n) >= 0, with equality if and only if n is a 5-full number (A069492) larger than 1. a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117). a(n) <= A000005(n), with equality if and only if n is cubefree (A004709). Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2. Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma - 1 - 6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). MATHEMATICA f[p_, e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] PROG (PARI) a(n) = vecprod(apply(x -> [2, 3, 2, 1, 0][min(x, 5)], factor(n)[, 2])); CROSSREFS Cf. A000005, A004709, A005117, A007427, A069492, A073184. Cf. A001620, A002117, A244115. Sequence in context: A359302 A366147 A367170 * A038148 A366742 A141829 Adjacent sequences: A369303 A369304 A369305 * A369307 A369308 A369309 KEYWORD nonn,easy,mult AUTHOR Amiram Eldar, Jan 19 2024 STATUS approved

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Last modified September 7 05:42 EDT 2024. Contains 375729 sequences. (Running on oeis4.)