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A360539 a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).

%I A360539 #17 Aug 06 2024 02:04:32
%S A360539 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,
%T A360539 28,29,30,31,1,33,34,35,36,37,38,39,5,41,42,43,44,45,46,47,3,49,50,51,
%U A360539 52,53,2,55,7,57,58,59,60,61,62,63,1,65,66,67,68,69,70
%N A360539 a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).
%C A360539 Equivalently, a(n) is the least divisor d of n such that n/d is a cubefull number (A036966).
%H A360539 Amiram Eldar, <a href="/A360539/b360539.txt">Table of n, a(n) for n = 1..10000</a>
%F A360539 a(n) = 1 if and only if n is a cubefull number (A036966).
%F A360539 a(n) = n if and only if n is a cubefree number (A004709).
%F A360539 a(n) >= A055231(n) with equality if and only if n is in A337050.
%F A360539 a(n) = n/A360540(n).
%F A360539 Multiplicative with a(p^e) = p^e if e <= 2, and 1 otherwise.
%F A360539 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)).
%F A360539 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... .
%t A360539 f[p_, e_] := If[e < 3, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360539 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2], 1));}
%o A360539 (Python)
%o A360539 from math import prod
%o A360539 from sympy import factorint
%o A360539 def A360539(n): return prod(p**e for p,e in factorint(n).items() if e<=2) # _Chai Wah Wu_, Aug 06 2024
%Y A360539 Cf. A004709, A007948, A036966, A055231, A360540.
%K A360539 nonn,easy,mult
%O A360539 1,2
%A A360539 _Amiram Eldar_, Feb 11 2023