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A374035 a(n) = gcd(A328845(n), A328846(n)), where A328845 and A328846 are the first and second Fibonacci-based variants of the arithmetic derivative.

%I A374035 #7 Jun 28 2024 10:04:28
%S A374035 0,0,1,1,4,5,1,1,12,6,5,1,4,1,3,5,32,1,3,1,20,1,3,1,4,50,1,27,8,1,5,1,
%T A374035 80,1,1,25,12,1,1,1,20,1,1,1,20,15,7,1,16,14,25,1,4,1,27,20,4,1,1,1,
%U A374035 80,1,33,3,192,10,1,1,8,1,5,1,12,1,1,25,12,1,1,1,80,108,1,1,4,15,1,1,4,1,15,1,60,1,1,10,16
%N A374035 a(n) = gcd(A328845(n), A328846(n)), where A328845 and A328846 are the first and second Fibonacci-based variants of the arithmetic derivative.
%H A374035 Antti Karttunen, <a href="/A374035/b374035.txt">Table of n, a(n) for n = 0..75025</a>
%H A374035 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A374035 For all n>= 0, a(5*n) == 0 (mod 5). [There are multiples of 5 at other positions also]
%o A374035 (PARI)
%o A374035 A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
%o A374035 A328846(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(2+primepi(f[i,1]))/f[i, 1]));
%o A374035 A374035(n) = gcd(A328845(n),A328846(n));
%Y A374035 Cf. A328845, A328846.
%Y A374035 Cf. A374036, A374037 (indices of even terms), A374038 (of odd terms).
%K A374035 nonn
%O A374035 0,5
%A A374035 _Antti Karttunen_, Jun 28 2024