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A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

%I A328846 #21 Jun 30 2024 09:03:34
%S A328846 0,0,2,3,8,5,12,8,24,18,20,13,36,21,30,30,64,34,54,55,60,45,48,89,96,
%T A328846 50,68,81,88,144,90,233,160,72,102,75,144,377,148,102,160,610,132,987,
%U A328846 140,135,224,1597,240,112,150,153,188,2584,216,120,232,222,346,4181,240,6765,528,198,384,170,210,10946,272,336,220,17711,360
%N A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.
%H A328846 Antti Karttunen, <a href="/A328846/b328846.txt">Table of n, a(n) for n = 0..10000</a>
%H A328846 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A328846 a(n) = n * Sum e_j * A000045(2+A000720(p_j))/p_j for n = Product p_j^e_j.
%F A328846 a(A000040(n)) = A000045(2+n).
%F A328846 A007895(a(n)) = A328848(n).
%o A328846 (PARI) 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]));
%Y A328846 Cf. A000040, A000045, A000720, A003965, A007895, A328848.
%Y A328846 Cf. also A003415, A258851, A328768, A328769, A328845 for other arithmetic derivatives, and also A371192 for another PrimePi-Fibonacci variant.
%Y A328846 Cf. A374035 [= gcd(a(n), A328845(n))], A374048 (antiparity of this sequence), A374049 (indices of even terms), A374050 (of odd terms).
%K A328846 nonn
%O A328846 0,3
%A A328846 _Antti Karttunen_, Oct 28 2019