A100107 Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..

1, 4, 5, 11, 12, 26, 30, 58, 81, 138, 200, 355, 522, 876, 1380, 2265, 3572, 5880, 9350, 15272, 24510, 39806, 64080, 104084, 167773, 271968, 439285, 711530, 1149852, 1862022, 3010350, 4873112, 7881400, 12755618, 20633280, 33391491, 54018522, 87413156

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

Cf. A000032, A007435, A100279.

Sequence in context: A027708 A047374 A241653 * A066828 A163098 A216562

Adjacent sequences: A100104 A100105 A100106 * A100108 A100109 A100110

Keyword

nonn

Author

Jonathan Vos Post, Dec 26 2004

Extensions

More terms from Emeric Deutsch, Jul 31 2005

Status

approved