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

%I A007435 M0631 #49 Apr 18 2025 21:21:08
%S A007435 1,2,3,5,6,12,14,26,37,62,90,159,234,392,618,1013,1598,2630,4182,6830,
%T A007435 10962,17802,28658,46548,75031,121628,196455,318206,514230,832722,
%U A007435 1346270,2179322,3524670,5704486,9227484,14933129,24157818,39092352,63246222,102341006
%N A007435 Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...
%C A007435 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 A007435 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007435 T. D. Noe, <a href="/A007435/b007435.txt">Table of n, a(n) for n = 1..1000</a>
%H A007435 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007435 Row sums of A051731 * A127647. - _Gary W. Adamson_, Jan 22 2007
%F A007435 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 A007435 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 A007435 a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, May 21 2018
%F A007435 From _Ridouane Oudra_, Apr 12 2025 : (Start)
%F A007435 a(n) = Sum_{d|n} Fibonacci(d).
%F A007435 a(n) = Sum_{d|n} mu(d)*A034772(n/d).
%F A007435 a(n) = A245282(n) - A108046(n).
%F A007435 a(n) = 2*A245282(n) - A100107(n).
%F A007435 a(n) = (A108031(n) + A108046(n))/2. (End)
%e A007435 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 + ...
%p A007435 with(numtheory): seq(add(combinat[fibonacci](d), d in divisors(n)), n=1..60); # _Ridouane Oudra_, Apr 12 2025
%t A007435 Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* _T. D. Noe_, Aug 14 2012 *)
%t A007435 a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* _Jean-François Alcover_, Dec 01 2015 *)
%o A007435 (PARI) {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* _Michael Somos_, Apr 15 2012 */
%Y A007435 Cf. A000045, A051731, A127647, A245282, A108046, A034772, A100107, A108031, A008683.
%K A007435 nonn,easy
%O A007435 1,2
%A A007435 _N. J. A. Sloane_
%E A007435 More terms from _Joerg Arndt_, Aug 14 2012