A100279 -id:A100279 - cp's OEIS frontend

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

Jonathan Vos Post, Dec 26 2004

nonn

			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.

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

Cf. A000032, A007435, A100279.

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

Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)

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

More terms from Emeric Deutsch, Jul 31 2005

A130878 Inverse Moebius transform of A100107.

Original entry on oeis.org

1, 5, 6, 16, 13, 36, 31, 74, 87, 155, 201, 402, 523, 911, 1398, 2339, 3573, 5997, 9351, 15438, 24546, 40011, 64081, 104544, 167786, 272495, 439372, 712452, 1149853, 1863588, 3010351, 4875451, 7881606, 12759195, 20633323, 33397854, 54018523, 87422511, 141423378

Offset: 1

R. J. Mathar, Aug 21 2007

Or, the inverse Moebius transform of the inverse Moebius transform of the Lucas numbers A000032.

Cf. A000032, A100107, A100279.

A000032 := proc(n) option remember; if n =0 then 2; elif n = 1 then 1; else A000032(n-1)+A000032(n-2) ; fi ; end: A100107 := proc(n) option remember ; local a,dvs,d ; a := 0: dvs := numtheory[divisors](n) ; for d in dvs do a := a+ A000032(d) ; od: RETURN(a) ; end: a := proc(n) local a,dvs,d ; a := 0: dvs := numtheory[divisors](n) ; for d in dvs do a := a+ A100107(d) ; od: RETURN(a) ; end: seq(a(n),n=1..100);
# second Maple program:
a:= ((p-> j-> add(p(d), d=numtheory[divisors](j)))@@2)
     (n-> (<<1|1>, <1|0>>^n.<<2, -1>>)[1, 1]):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 23 2024

A100107[n_] := LucasL /@ Divisors[n] // Total;
a[n_] := A100107 /@ Divisors[n] // Total;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 23 2024 *)

a(n) = Sum_{d|n} A100107(d).

cp's OEIS Frontend

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

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