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A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

%I A349141 #15 Nov 15 2021 03:40:11
%S A349141 0,1,1,6,1,9,1,26,9,13,1,44,1,17,15,98,1,57,1,68,19,25,1,176,15,29,57,
%T A349141 92,1,105,1,342,27,37,23,252,1,41,31,280,1,141,1,140,111,49,1,636,21,
%U A349141 125,39,164,1,309,31,384,43,61,1,480,1,65,147,1138,35,213,1,212,51,209,1,960,1,77,155,236,35,249,1,1028
%N A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).
%C A349141 Dirichlet convolution of Euler phi (A000010) with A348507.
%C A349141 Möbius transform of A349140.
%H A349141 Antti Karttunen, <a href="/A349141/b349141.txt">Table of n, a(n) for n = 1..16384</a>
%F A349141 a(n) = Sum_{d|n} A000010(n/d) * A348507(d).
%F A349141 a(n) = Sum_{d|n} A008683(n/d) * A349140(d).
%F A349141 a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
%F A349141 For all n >= 1, a(n) >= A347131(n) >= A348981(n).
%F A349141 a(n) = A349171(n) - A018804(n). - _Antti Karttunen_, Nov 14 2021
%t A349141 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 A349141 (PARI)
%o A349141 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349141 A348507(n) = (A003959(n) - n);
%o A349141 A349141(n) = sumdiv(n,d,eulerphi(d)*A348507(n/d));
%Y A349141 Cf. A000010, A003959, A008683, A018804, A348507, A349140 (inverse Möbius transform), A349142, A349143, A349171.
%Y A349141 Cf. also A347131, A348981.
%K A349141 nonn
%O A349141 1,4
%A A349141 _Antti Karttunen_, Nov 08 2021