OFFSET
1,2
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = Sum_{d|n} Lucas(d) = Sum_{d|n} A000032(d).
G.f.: Sum_{k>=1} Lucas(k) * x^k/(1 - x^k) = Sum_{k>=1} x^k * (1 + 2*x^k)/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Aug 14 2019
a(n) ~ phi^n, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 10 2021
EXAMPLE
a(2) = 4 because the prime 2 is divisible only by 1 and 2, so L(1) + L(2) = 1 + 3 = 4.
a(3) = 5 because the prime 3 is divisible only by 1 and 3, so L(1) + L(3) = 1 + 4 = 5.
a(4) = 11 because the semiprime 4 is divisible only by 1, 2, 4, so L(1) + L(2) + L(4) = 1 + 3 + 7 = 11.
MAPLE
with(numtheory): with(combinat): a:=proc(n) local div: div:=divisors(n): sum(2*fibonacci(div[j]+1)-fibonacci(div[j]), j=1..tau(n)) end: seq(a(n), n=1..42); # Emeric Deutsch, Jul 31 2005
MATHEMATICA
Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Dec 26 2004
EXTENSIONS
More terms from Emeric Deutsch, Jul 31 2005
STATUS
approved