A084907 Indices of Fibonacci numbers that satisfy: Sum_{k>=1} 1/F(a(k)) = 1, where F(k) gives the k-th Fibonacci number with F(0)=1, such that the partial sums are nearest to, but never exceed, unity.

2, 3, 5, 8, 10, 15, 26, 31, 33, 36, 40, 42, 47, 49, 51, 53, 56, 58, 61, 77, 79, 81, 84, 88, 90, 92, 95, 97, 99, 102, 104, 106, 111, 113, 115, 120, 122, 127, 129, 132, 136, 138, 141, 145, 147, 150, 157, 159, 162, 164, 172, 177, 181, 183, 185, 200, 203, 206, 209, 211

Offset

1,1

Comments

Corresponding Fibonacci numbers are given by A084908.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 29, 292, 2811, 27695, ... Apparently, the asymptotic density of this sequence is 1/(sqrt(5)*phi) = 0.27639... (A244847). - Amiram Eldar, Feb 15 2022

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

1 = 1/F(2) + 1/F(3) + 1/F(5) + 1/F(8) + 1/F(10) + 1/F(15) + ... = 1/2 + 1/3 + 1/8 + 1/34 + 1/89 + 1/987 + 1/196418 + 1/2178309 +...

Mathematica

seq = {}; s = 1; m = 3; Do[AppendTo[seq, m - 1]; s -= (1/Fibonacci[m]); While[Fibonacci[m] <= 1/s, m++], {60}]; seq (* Amiram Eldar, Nov 01 2019 *)

Prog

(PARI) x=1; a=2; S=0; for(n=1, 100, b=a+1; while(abs(S+1/fibonacci(b))>x, b++); S=S+1/fibonacci(b); a=b; print1(b-1, ", "))

Crossrefs

Cf. A000045, A084908, A084909, A244847.

Sequence in context: A306972 A087279 A246346 * A158724 A289276 A181100

Adjacent sequences: A084904 A084905 A084906 * A084908 A084909 A084910

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2003

Extensions

Terms a(41) onward corrected by Amiram Eldar, Nov 01 2019

Status

approved