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A348498 a(n) = gcd(A003415(n), A347130(n)) / A003557(n), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

%I A348498 #10 Feb 02 2022 20:36:40
%S A348498 0,1,1,1,1,5,1,3,1,7,1,8,1,9,8,2,1,7,1,12,10,13,1,11,1,15,3,16,1,31,1,
%T A348498 5,14,19,12,5,1,21,16,17,1,41,1,24,13,25,1,14,1,9,20,28,1,9,16,23,22,
%U A348498 31,1,46,1,33,17,3,18,61,1,36,26,59,1,13,1,39,11,40,18,71,1,22,2,43,1,62,22,45,32,35,1,41
%N A348498 a(n) = gcd(A003415(n), A347130(n)) / A003557(n), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.
%H A348498 Antti Karttunen, <a href="/A348498/b348498.txt">Table of n, a(n) for n = 1..20000</a>
%F A348498 a(n) = A348497(n) / A003557(n).
%F A348498 a(n) = gcd(A342001(n), A347129(n)).
%t A348498 f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Table[GCD[f[n], DivisorSum[n, # f[n/#] &]]*Apply[Times, FactorInteger[n][[All, 1]]]/n, {n, 90}] (* _Michael De Vlieger_, Oct 25 2021 *)
%o A348498 (PARI)
%o A348498 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A348498 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A348498 A347130(n) = sumdiv(n,d,d*A003415(n/d));
%o A348498 A348498(n) = (gcd(A003415(n), A347130(n))/A003557(n));
%Y A348498 Cf. A003415, A003557, A342001, A347129, A347130, A348497, A348502 [= a(A276086(n))].
%Y A348498 Cf. also A348494, A348496.
%K A348498 nonn
%O A348498 1,6
%A A348498 _Antti Karttunen_, Oct 23 2021