cp's OEIS Frontend

A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

%I A382660 #8 Apr 02 2025 12:43:50
%S A382660 1,1,2,4,2,6,7,4,10,12,6,8,16,18,12,10,22,14,12,26,28,8,30,31,20,16,
%T A382660 24,36,18,24,28,40,12,42,22,46,32,52,26,40,42,36,28,58,60,30,48,20,66,
%U A382660 44,24,70,72,36,60,24,78,40,82,64,42,56,70,88,72,60,46,72
%N A382660 The unitary totient function applied to the exponentially odd numbers (A268335).
%H A382660 Amiram Eldar, <a href="/A382660/b382660.txt">Table of n, a(n) for n = 1..10000</a>
%F A382660 a(n) = A047994(A268335(n)).
%F A382660 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
%t A382660 f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uphi /@ Select[Range[100], expOddQ]
%o A382660 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
%o A382660 isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
%o A382660 list(lim) = apply(uphi, select(isexpodd, vector(lim, i, i)));
%Y A382660 Cf. A013662, A047994, A065463, A268335.
%Y A382660 Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382661.
%K A382660 nonn,easy
%O A382660 1,3
%A A382660 _Amiram Eldar_, Apr 02 2025