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%I #10 Nov 24 2021 18:19:20
%S 0,1,1,2,1,0,1,6,3,0,1,2,1,0,0,14,1,6,1,2,0,0,1,2,5,0,15,2,1,0,1,34,0,
%T 0,0,18,1,0,0,2,1,0,1,2,6,0,1,6,7,20,0,2,1,6,0,2,0,0,1,0,1,0,6,78,0,0,
%U 1,2,0,0,1,42,1,0,20,2,0,0,1,6,51,0,1,0,0,0,0,2,1,0,0,2,0,0,0,10,1,42,6,50,1,0,1,2,0
%N Dirichlet convolution of arithmetic derivative with A325126 [Dirichlet inverse of rad(n)].
%H Antti Karttunen, <a href="/A349618/b349618.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) = Sum_{d|n} A003415(d) * A325126(n/d).
%t f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 23 2021 *)
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
%o memoA325126 = Map();
%o A325126(n) = if(1==n,1,my(v); if(mapisdefined(memoA325126,n,&v), v, v = -sumdiv(n,d,if(d<n,A007947(n/d)*A325126(d),0)); mapput(memoA325126,n,v); (v)));
%o A349618(n) = sumdiv(n,d,A003415(d)*A325126(n/d));
%Y Cf. A003415, A007947, A325126.
%Y Cf. also A349394, A349612.
%K nonn
%O 1,4
%A _Antti Karttunen_, Nov 23 2021
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