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A007435
Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...
14
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
1,2
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
N. J. A. Sloane, Transforms
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)
EXAMPLE
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 + ...
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 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */
KEYWORD
nonn,easy
EXTENSIONS
More terms from Joerg Arndt, Aug 14 2012
STATUS
approved