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A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).

%I A382663 #8 Apr 02 2025 12:43:56
%S A382663 1,3,8,15,24,24,48,80,72,120,120,168,144,192,288,240,360,360,384,360,
%T A382663 528,624,504,720,840,576,960,960,864,1152,1200,1368,1080,1344,1680,
%U A382663 1152,1848,1800,1920,1584,2208,2400,1872,2304,2520,2808,2880,2880,2520,3480,2880
%N A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).
%H A382663 Amiram Eldar, <a href="/A382663/b382663.txt">Table of n, a(n) for n = 1..10000</a>
%F A382663 a(n) = A191414(A004709(n)).
%F A382663 Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)^3/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4 - 1/p^6 + 1/p^7) = 0.42656661743049439763... .
%t A382663 f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uj2 /@ Select[Range[100], cubeFreeQ]
%o A382663 (PARI) uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
%o A382663 iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
%o A382663 list(lim) = apply(uj2, select(iscubefree, vector(lim, i, i)));
%Y A382663 Cf. A002117, A004709, A191414.
%Y A382663 Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382662.
%K A382663 nonn,easy
%O A382663 1,2
%A A382663 _Amiram Eldar_, Apr 02 2025