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 A100107 Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,.. 5

%I

%S 1,4,5,11,12,26,30,58,81,138,200,355,522,876,1380,2265,3572,5880,9350,

%T 15272,24510,39806,64080,104084,167773,271968,439285,711530,1149852,

%U 1862022,3010350,4873112,7881400,12755618,20633280,33391491,54018522,87413156

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

%H T. D. Noe, <a href="/A100107/b100107.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{d|n} Lucas(d) = Sum_{d|n} A000032(d).

%F 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

%e a(2) = 4 because the prime 2 is divisible only by 1 and 2, so L(1) + L(2) = 1 + 3 = 4.

%e a(3) = 5 because the prime 3 is divisible only by 1 and 3, so L(1) + L(3) = 1 + 4 = 5.

%e 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.

%p 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

%t Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* _T. D. Noe_, Aug 14 2012 *)

%Y Cf. A000032, A007435, A100279.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Dec 26 2004

%E More terms from _Emeric Deutsch_, Jul 31 2005

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Last modified July 5 07:37 EDT 2020. Contains 335462 sequences. (Running on oeis4.)