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A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

%I A349132 #14 Nov 14 2021 10:14:02
%S A349132 1,4,6,10,10,24,14,22,24,40,22,60,26,56,60,46,34,96,38,100,84,88,46,
%T A349132 132,70,104,84,140,58,240,62,94,132,136,140,240,74,152,156,220,82,336,
%U A349132 86,220,240,184,94,276,140,280,204,260,106,336,220,308,228,232,118,600,122,248,336,190,260,528,134,340,276,560
%N A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.
%C A349132 Dirichlet convolution of A003958 with Dedekind psi function, A001615.
%H A349132 Antti Karttunen, <a href="/A349132/b349132.txt">Table of n, a(n) for n = 1..20000</a>
%F A349132 a(n) = Sum_{d|n} A001615(d) * A003958(n/d).
%F A349132 a(n) = A327251(n) - A348982(n).
%F A349132 For all n >= 1, a(n) <= A349172(n).
%F A349132 Multiplicative with a(p^e) = (p+1)*p^e - p*(p-1)^e. - _Amiram Eldar_, Nov 09 2021
%t A349132 f[p_, e_] := (p + 1)*p^e - p*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *)
%o A349132 (PARI)
%o A349132 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
%o A349132 A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A349132 A349132(n) = sumdiv(n,d,A001615(d)*A003958(n/d));
%Y A349132 Cf. A001615, A003958, A327251, A348982, A349130, A349131, A349133, A349172.
%K A349132 nonn,mult
%O A349132 1,2
%A A349132 _Antti Karttunen_, Nov 09 2021