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A360539
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a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).
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9
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1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70
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OFFSET
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1,2
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COMMENTS
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Equivalently, a(n) is the least divisor d of n such that n/d is a cubefull number (A036966).
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LINKS
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FORMULA
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a(n) = 1 if and only if n is a cubefull number (A036966).
a(n) = n if and only if n is a cubefree number (A004709).
Multiplicative with a(p^e) = p^e if e <= 2, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(-3*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 0.4213813264... .
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MATHEMATICA
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f[p_, e_] := If[e < 3, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2], 1)); }
(Python)
from math import prod
from sympy import factorint
def A360539(n): return prod(p**e for p, e in factorint(n).items() if e<=2) # Chai Wah Wu, Aug 06 2024
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CROSSREFS
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KEYWORD
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nonn,easy,mult,changed
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AUTHOR
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STATUS
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approved
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