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A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

%I A348981 #13 Nov 15 2021 03:40:35
%S A348981 0,1,1,4,1,7,1,12,7,11,1,24,1,15,13,32,1,35,1,40,17,23,1,68,13,27,35,
%T A348981 56,1,71,1,80,25,35,21,112,1,39,29,116,1,99,1,88,77,47,1,176,19,91,37,
%U A348981 104,1,151,29,164,41,59,1,232,1,63,105,192,33,155,1,136,49,159,1,308,1,75,117,152,33,183,1,304,151
%N A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).
%C A348981 Dirichlet convolution of Euler phi (A000010) with A322582.
%C A348981 Möbius transform of A348980.
%H A348981 Antti Karttunen, <a href="/A348981/b348981.txt">Table of n, a(n) for n = 1..16384</a>
%F A348981 a(n) = Sum_{d|n} A000010(n/d) * A322582(d).
%F A348981 a(n) = Sum_{d|n} A008683(n/d) * A348980(d).
%F A348981 a(n) = Sum_{k=1..n} A322582(gcd(n,k)).
%F A348981 For all n >= 1, a(n) <= A347131(n) <= A349141(n).
%F A348981 a(n) = A018804(n) - A349131(n). - _Antti Karttunen_, Nov 14 2021
%t A348981 f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#]) * EulerPhi[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)
%o A348981 (PARI)
%o A348981 A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A348981 A322582(n) = (n-A003958(n));
%o A348981 A348981(n) = sumdiv(n,d,A322582(n/d)*eulerphi(d));
%Y A348981 Cf. A000010, A003958, A008683, A018804, A322582, A348980 (Inverse Möbius transform), A348981, A348982, A348983, A349131.
%Y A348981 Cf. also A347131, A349141.
%K A348981 nonn
%O A348981 1,4
%A A348981 _Antti Karttunen_, Nov 08 2021