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A349133 Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

%I A349133 #11 Nov 14 2021 17:44:34
%S A349133 0,1,1,5,1,8,1,17,8,12,1,32,1,16,14,49,1,43,1,52,18,24,1,100,14,28,43,
%T A349133 72,1,87,1,129,26,36,22,151,1,40,30,168,1,119,1,112,91,48,1,276,20,
%U A349133 103,38,132,1,194,30,236,42,60,1,323,1,64,123,321,34,183,1,172,50,183,1,443,1,76,131,192,34,215,1,472
%N A349133 Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).
%H A349133 Antti Karttunen, <a href="/A349133/b349133.txt">Table of n, a(n) for n = 1..20000</a>
%F A349133 a(n) = Sum_{d|n} A003415(d) * A003958(n/d).
%F A349133 For all n >= 1, a(n) <= A349173(n).
%t A349133 f1[p_, e_] := e/p; f2[p_, e_] := (p - 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *)
%o A349133 (PARI)
%o A349133 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349133 A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A349133 A349133(n) = sumdiv(n,d,A003415(d)*A003958(n/d));
%Y A349133 Cf. A003415, A003958, A349129, A349130, A349131, A349132, A349173.
%K A349133 nonn
%O A349133 1,4
%A A349133 _Antti Karttunen_, Nov 09 2021