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A371192
A variant of the arithmetic derivative with a(prime(k)) = Fibonacci(k) and a(u*v) = a(u)*v + u*a(v).
3
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
0,5
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.
EXAMPLE
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.
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Bedard, Mar 14 2024
STATUS
approved