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A366126 The largest unitary divisor of n that is a cube.

%I A366126 #7 Sep 30 2023 21:57:37
%S A366126 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,27,1,1,1,1,1,1,1,
%T A366126 1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,27,1,8,1,1,1,1,1,1,1,64,1,1,1,
%U A366126 1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A366126 The largest unitary divisor of n that is a cube.
%C A366126 First differs from A056191 at n = 32.
%H A366126 Amiram Eldar, <a href="/A366126/b366126.txt">Table of n, a(n) for n = 1..10000</a>
%F A366126 Multiplicative with a(p^e) = p^e if e is divisible by 3 and 1 otherwise.
%F A366126 a(n) = n if and only if n is a positive cube (A000578 \ {0}).
%F A366126 A001221(a(n)) = A366124(n).
%F A366126 A001222(a(n)) = A366125(n).
%F A366126 Sum_{k=1..n} a(k) ~ c * n^(4/3), where c = (1/4) * Product_{p prime} (1 + (p^(1/3) + p^(5/3))/(1 + p + p^2 + p^3)) = 0.61488587249270755696... .
%t A366126 f[p_, e_] := If[Divisible[e, 3], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366126 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] % 3, 1, f[i,1]^f[i,2]));}
%Y A366126 Cf. A001221, A001222, A000578, A056191, A350388, A366124, A366125.
%K A366126 nonn,easy,mult
%O A366126 1,8
%A A366126 _Amiram Eldar_, Sep 30 2023