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A348497 a(n) = gcd(A003415(n), A347130(n)), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

%I A348497 #9 Feb 02 2022 20:36:34
%S A348497 0,1,1,2,1,5,1,12,3,7,1,16,1,9,8,16,1,21,1,24,10,13,1,44,5,15,27,32,1,
%T A348497 31,1,80,14,19,12,30,1,21,16,68,1,41,1,48,39,25,1,112,7,45,20,56,1,81,
%U A348497 16,92,22,31,1,92,1,33,51,96,18,61,1,72,26,59,1,156,1,39,55,80,18,71,1,176,54,43,1,124,22,45
%N A348497 a(n) = gcd(A003415(n), A347130(n)), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.
%H A348497 Antti Karttunen, <a href="/A348497/b348497.txt">Table of n, a(n) for n = 1..20000</a>
%F A348497 a(n) = gcd(A003415(n), A347130(n)).
%F A348497 a(n) = A003557(n) * A348498(n).
%t A348497 f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Table[GCD[f[n], DivisorSum[n, # f[n/#] &]], {n, 86}] (* _Michael De Vlieger_, Oct 25 2021 *)
%o A348497 (PARI)
%o A348497 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A348497 A347130(n) = sumdiv(n,d,d*A003415(n/d));
%o A348497 A348497(n) = gcd(A003415(n), A347130(n));
%Y A348497 Cf. A003415, A003557, A347130, A348498.
%Y A348497 Cf. also A348492, A348495.
%K A348497 nonn
%O A348497 1,4
%A A348497 _Antti Karttunen_, Oct 23 2021