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