A371192 A variant of the arithmetic derivative with a(prime(k)) = Fibonacci(k) and a(u*v) = a(u)*v + u*a(v).

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, 27, 40, 55, 37, 89, 80, 26, 43, 29, 60, 144, 61, 37, 76, 233, 53, 377, 64, 48, 91, 610, 112, 42, 65, 56, 84, 987, 81, 47, 108, 82, 139, 1597, 104, 2584, 209, 69, 192

Offset: 0

Author

Paul Bedard, Mar 14 2024

Keywords

• nonn

Examples

			a(1)=0 is implied by a(q*r)=q*a(r)+r*a(q).
a(2)=1 since 2 = prime(k) for k=1, and the corresponding Fibonacci number is Fibonacci(k) = 1.
a(4) = a(2*2) = 2*a(2)+2*a(2) = 4.
		

Crossrefs

Cf. A000040, A000045, A000720, A003415, A328845, A328846.

Programs

Maple

with(numtheory): F:=combinat[fibonacci]:
a:= n-> n*add(i[2]*F(pi(i[1]))/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..64);  # Alois P. Heinz, Mar 20 2024

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

Formula

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.