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A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

%I A383954 #16 Aug 19 2025 16:37:23
%S A383954 1,0,1,1,3,0,5,3,5,0,9,1,11,0,3,7,15,0,17,3,5,0,21,3,19,0,17,5,27,0,
%T A383954 29,15,9,0,15,5,35,0,11,9,39,0,41,9,15,0,45,7,41,0,15,11,51,0,27,15,
%U A383954 17,0,57,3,59,0,25,31,33,0,65,15,21,0,69,15,71,0,19,17,45,0,77,21
%N A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.
%C A383954 This is the phi- function in Sandor and Atanassof.
%H A383954 Paolo Xausa, <a href="/A383954/b383954.txt">Table of n, a(n) for n = 1..10000</a>
%H A383954 József Sándor and Krassimir Atanassov, <a href="https://doi.org/10.7546/nntdm.2024.30.4.851-856">Some new arithmetic functions</a>, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856.
%F A383954 From _Amiram Eldar_, Aug 19 2025: (Start)
%F A383954 Multiplicative with a(p^e) = (p-1)*p^(e-1) - 1.
%F A383954 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
%F A383954 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p+1) - (p+1)/p^2) = 0.39439177573628632634... . (End)
%t A383954 A383954[n_] := If[n == 1, 1, Times @@ (EulerPhi[Power @@@ FactorInteger[n]] - 1)];
%t A383954 Array[A383954, 100] (* _Paolo Xausa_, Aug 19 2025 *)
%o A383954 (PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, p=f[k,1]; eulerphi(f[k,1]^f[k,2])-1);
%Y A383954 Cf. A000010 (phi), A107758 (sigma+), A057723 (sigma-), A055653 (phi+).
%K A383954 nonn,mult,easy
%O A383954 1,5
%A A383954 _Michel Marcus_, Aug 19 2025