%I #14 Jul 08 2024 13:59:32

%S 0,0,1,2,4,5,7,13,12,12,15,89,20,233,33,25,32,1597,33,4181,40,53,189,

%T 28657,52,50,479,54,80,514229,65,1346269,80,289,3211,100,84,24157817,

%U 8381,725,100,165580141,127,433494437,400,105,57337,2971215073,128,182,125,4825,984,53316291173,135,500,188,12581,1028487,956722026041,160

%N The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

%H Antti Karttunen, <a href="/A328845/b328845.txt">Table of n, a(n) for n = 0..1001</a>

%F a(n) = n * Sum e_j * A000045(p_j)/p_j for n = Product p_j^e_j.

%F a(A000040(n)) = A030426(n).

%F A007895(a(n)) = A328847(n).

%o (PARI) 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]));

%Y Cf. A000040, A000045, A007895, A030426, A113175, A328847, A374201.

%Y Cf. A374046 (indices of even terms), A374047 (of odd terms), A374122 (of multiples of 3), A374202 (2-adic valuation), A374203 (3-adic valuation), A374205 (5-adic valuation), A374125 [a(n) mod 360].

%Y Cf. A374106 [gcd(a(n), A113177(n))], A374035 [gcd(a(n), A328846(n))], A374116 [gcd(a(n), A328768(n))].

%Y For variants of the same formula, see A003415, A258851, A328768, A328769, A328846, A371192.

%K nonn

%O 0,4

%A _Antti Karttunen_, Oct 28 2019