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

Author

Jonathan Vos Post, Dec 26 2004

Keywords

• nonn

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000032, A007435, A100279.

Programs

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 *)

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

Extensions

More terms from Emeric Deutsch, Jul 31 2005