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A328845 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.

%I A328845 #19 Dec 16 2024 05:12:01
%S A328845 0,0,1,2,4,5,7,13,12,12,15,89,20,233,33,25,32,1597,33,4181,40,53,189,
%T A328845 28657,52,50,479,54,80,514229,65,1346269,80,289,3211,100,84,24157817,
%U A328845 8381,725,100,165580141,127,433494437,400,105,57337,2971215073,128,182,125,4825,984,53316291173,135,500,188,12581,1028487,956722026041,160
%N A328845 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 A328845 Antti Karttunen, <a href="/A328845/b328845.txt">Table of n, a(n) for n = 0..1001</a>
%F A328845 a(n) = n * Sum e_j * A000045(p_j)/p_j for n = Product p_j^e_j.
%F A328845 a(A000040(n)) = A030426(n).
%F A328845 A007895(a(n)) = A328847(n).
%t A328845 A328845[n_] := If[n <= 1, 0, n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]]];
%t A328845 Array[A328845, 100, 0] (* _Paolo Xausa_, Dec 16 2024 *)
%o A328845 (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 A328845 Cf. A000040, A000045, A007895, A030426, A113175, A328847, A374201.
%Y A328845 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 A328845 Cf. A374106 [gcd(a(n), A113177(n))], A374035 [gcd(a(n), A328846(n))], A374116 [gcd(a(n), A328768(n))].
%Y A328845 For variants of the same formula, see A003415, A258851, A328768, A328769, A328846, A371192.
%K A328845 nonn
%O A328845 0,4
%A A328845 _Antti Karttunen_, Oct 28 2019