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A382662 The unitary totient function applied to the cubefree numbers (A004709).

%I A382662 #7 Apr 02 2025 12:44:01
%S A382662 1,1,2,3,4,2,6,8,4,10,6,12,6,8,16,8,18,12,12,10,22,24,12,18,28,8,30,
%T A382662 20,16,24,24,36,18,24,40,12,42,30,32,22,46,48,24,32,36,52,40,36,28,58,
%U A382662 24,60,30,48,48,20,66,48,44,24,70,72,36,48,54,60,24,78,40
%N A382662 The unitary totient function applied to the cubefree numbers (A004709).
%H A382662 Amiram Eldar, <a href="/A382662/b382662.txt">Table of n, a(n) for n = 1..10000</a>
%F A382662 a(n) = A047994(A004709(n)).
%F A382662 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)^2/2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.41625329674394407438... .
%t A382662 f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uphi /@ Select[Range[100], cubeFreeQ]
%o A382662 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1); }
%o A382662 iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
%o A382662 list(lim) = apply(uphi, select(iscubefree, vector(lim, i, i)));
%Y A382662 Cf. A002117, A004709, A047994.
%Y A382662 Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382663.
%K A382662 nonn,easy
%O A382662 1,3
%A A382662 _Amiram Eldar_, Apr 02 2025