A007435 Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...

1, 2, 3, 5, 6, 12, 14, 26, 37, 62, 90, 159, 234, 392, 618, 1013, 1598, 2630, 4182, 6830, 10962, 17802, 28658, 46548, 75031, 121628, 196455, 318206, 514230, 832722, 1346270, 2179322, 3524670, 5704486, 9227484, 14933129, 24157818, 39092352, 63246222, 102341006

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For p prime, a(p) == k (mod p) where k = 0 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 1 if p = 5. - Michael Somos, Apr 15 2012

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 12*x^6 + 14*x^7 + 26*x^8 + 37*x^9 + 62*x^10 + ...
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000045, A051731, A127647, A245282, A108046, A034772, A100107, A108031, A008683.

Programs

Maple

with(numtheory): seq(add(combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 12 2025

Mathematica

Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */

Formula

Row sums of A051731 * A127647. - Gary W. Adamson, Jan 22 2007

G.f.: Sum_{k>0} Fibonacci(k)*x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k-x^(2*k)). - Vladeta Jovovic, Dec 17 2002

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(Fibonacci(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018

a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2018

From Ridouane Oudra, Apr 12 2025 : (Start)

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

a(n) = Sum_{d|n} mu(d)*A034772(n/d).

a(n) = A245282(n) - A108046(n).

a(n) = 2*A245282(n) - A100107(n).

a(n) = (A108031(n) + A108046(n))/2. (End)

Extensions

More terms from Joerg Arndt, Aug 14 2012