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%I #8 Nov 06 2021 20:13:46
%S 1,3,4,1,6,1,8,1,1,1,12,4,14,1,1,3,18,3,20,2,1,1,24,4,1,1,2,12,30,1,
%T 32,1,1,1,1,48,38,1,1,54,42,1,44,4,12,1,48,4,1,1,1,18,54,3,1,4,1,1,60,
%U 8,62,1,2,1,1,1,68,2,1,3,72,12,74,1,2,12,1,1,80,2,1,1,84,16,1,1,1,12,90,3,1,4,1,1,1,4,98
%N a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.
%F a(n) = gcd(A003959(n), A129283(n)) = gcd(A003959(n), n+A003415(n)).
%F a(n) = gcd(A003959(n), A348970(n)) = gcd(A129283(n), A348970(n)).
%F a(n) = A129283(n) / A348973(n) = A003959(n) / A348974(n).
%t f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n])), Times @@ f2 @@@ f]; Array[a, 100] (* _Amiram Eldar_, Nov 06 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 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348972(n) = gcd(A003959(n),(n+A003415(n)));
%Y Cf. A003415, A003959, A129283, A348970, A348973, A348974.
%Y Cf. also A343226.
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 06 2021
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