A300895 - OEIS

%I #7 May 21 2018 15:47:26

%S 1,1,4,-3,6,-2,1,5,4,-4,1,-6,14,1,9,-11,1,-2,1,-8,25,1,1,2,6,-12,4,-3,

%T 1,-7,1,-11,4,35,6,-6,1,1,17,0,1,-23,1,-3,9,1,1,-14,1,-4,4,-16,1,-2,

%U 61,5,4,1,1,-11,1,1,25,-11,19,-2,1,-37,4,-4,1,2,1,1,9,-3,1,-15,1,-16,4,1,1,-27,6

%N L.g.f.: log(Product_{k>=2} (1 + x^Fibonacci(k))) = Sum_{n>=1} a(n)*x^n/n.

%F G.f.: Sum_{k>=2} Fibonacci(k)*x^Fibonacci(k)/(1 + x^Fibonacci(k)).

%F a(n) = n + 1 if n is an odd prime Fibonacci number (A005478 except a(1) = 2).

%e L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 - 2*x^6/6 + x^7/7 + 5*x^8/8 + 4*x^9/9 - 4*x^10/10 + ...

%e exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + 2*x^10 + ... + A000119(n)*x^n + ...

%t nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^Fibonacci[k]), {k, 2, 14}]], {x, 0, nmax}],x] Range[0, nmax]]

%t nmax = 85; Rest[CoefficientList[Series[Sum[Fibonacci[k] x^Fibonacci[k]/(1 + x^Fibonacci[k]), {k, 2, 14}], {x, 0, nmax}], x]]

%t Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(5 #^2 + 4)^(1/2)] || IntegerQ[(5 #^2 - 4)^(1/2)] &], {n, 85}]

%Y Cf. A000045, A000119, A005092, A005478.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, Mar 14 2018