A007435 - OEIS

%I M0631 #40 Jan 03 2024 03:54:23

%S 1,2,3,5,6,12,14,26,37,62,90,159,234,392,618,1013,1598,2630,4182,6830,

%T 10962,17802,28658,46548,75031,121628,196455,318206,514230,832722,

%U 1346270,2179322,3524670,5704486,9227484,14933129,24157818,39092352,63246222,102341006

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

%C 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

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

%H T. D. Noe, <a href="/A007435/b007435.txt">Table of n, a(n) for n = 1..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Row sums of A051731 * A127647. - _Gary W. Adamson_, Jan 22 2007

%F 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

%F 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

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

%e 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 + ...

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

%t a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* _Jean-François Alcover_, Dec 01 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* _Michael Somos_, Apr 15 2012 */

%Y Cf. A000045, A051731, A127647, A245282.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Joerg Arndt_, Aug 14 2012