A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A004709, A036966, A057521, A360539.

Sequence in context: A366906 A056201 A368329 * A088842 A284098 A010152

Adjacent sequences: A360537 A360538 A360539 * A360541 A360542 A360543

Keyword

nonn,easy,mult

Author

Amiram Eldar, Feb 11 2023

Status

approved