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A360540
a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).
10
1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1
OFFSET
1,8
LINKS
FORMULA
a(n) = 1 if and only if n is a cubefree number (A004709).
a(n) = n if and only if n is a cubefull number (A036966).
a(n) <= A057521(n) with equality if and only if n is in A337050.
a(n) = n/A360539(n).
Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - p^(1-s) + p^(-s) - p^(1-3*s) - p^(1-2*s) + p^(-2*s) + p^(3-3*s)).
MATHEMATICA
f[p_, e_] := If[e > 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 2, f[i, 1]^f[i, 2], 1)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Feb 11 2023
STATUS
approved