A328846 - OEIS

%I #21 Jun 30 2024 09:03:34

%S 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 50,68,81,88,144,90,233,160,72,102,75,144,377,148,102,160,610,132,987,

%U 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 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 Antti Karttunen, <a href="/A328846/b328846.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>

%F a(n) = n * Sum e_j * A000045(2+A000720(p_j))/p_j for n = Product p_j^e_j.

%F a(A000040(n)) = A000045(2+n).

%F A007895(a(n)) = A328848(n).

%o (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 Cf. A000040, A000045, A000720, A003965, A007895, A328848.

%Y Cf. also A003415, A258851, A328768, A328769, A328845 for other arithmetic derivatives, and also A371192 for another PrimePi-Fibonacci variant.

%Y Cf. A374035 [= gcd(a(n), A328845(n))], A374048 (antiparity of this sequence), A374049 (indices of even terms), A374050 (of odd terms).

%K nonn

%O 0,3

%A _Antti Karttunen_, Oct 28 2019