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%I #30 Apr 06 2024 20:54:19
%S 0,0,1,1,4,2,5,3,12,6,9,5,16,8,13,11,32,13,21,21,28,16,21,34,44,20,29,
%T 27,40,55,37,89,80,26,43,29,60,144,61,37,76,233,53,377,64,48,91,610,
%U 112,42,65,56,84,987,81,47,108,82,139,1597,104,2584,209,69,192
%N A variant of the arithmetic derivative with a(prime(k)) = Fibonacci(k) and a(u*v) = a(u)*v + u*a(v).
%F a(n) = n * Sum_{i=1..k} e[i] * Fibonacci(pi(p[i])) / p[i], where the prime factorization of n is n = Product_{i=1..k} p[i]^e[i], and pi(p) is the prime index prime(pi(p)) = p.
%e a(1)=0 is implied by a(q*r)=q*a(r)+r*a(q).
%e a(2)=1 since 2 = prime(k) for k=1, and the corresponding Fibonacci number is Fibonacci(k) = 1.
%e a(4) = a(2*2) = 2*a(2)+2*a(2) = 4.
%p with(numtheory): F:=combinat[fibonacci]:
%p a:= n-> n*add(i[2]*F(pi(i[1]))/i[1], i=ifactors(n)[2]):
%p seq(a(n), n=0..64); # _Alois P. Heinz_, Mar 20 2024
%o (PARI) a(n) = if(n==0,0, my(f=factor(n)); n*sum(k=1, #f~, f[k,2]*fibonacci(primepi(f[k,1]))/f[k,1])); \\ _Michel Marcus_, Mar 25 2024
%Y Cf. A000040, A000045, A000720, A003415, A328845, A328846.
%K nonn
%O 0,5
%A _Paul Bedard_, Mar 14 2024
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