A139095 Fibonacci numbers whose sum of proper divisors is also a Fibonacci number.

1, 1, 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241

Offset: 1

Author

Omar E. Pol, May 11 2008

Keywords

• nonn

Comments

Fibonacci numbers k such that A001065(k) is a Fibonacci number.

A001065(a(n)) is a Fibonacci number.

Certainly this contains 1 and the terms of A005478. Does it contain any other terms? - R. J. Mathar, Sep 17 2009

The next term, Fibonacci(359) = 4.754...*10^74, is too large to include in the data section. There are no composite Fibonacci numbers below A000045(1423) in this sequence. - Amiram Eldar, Mar 11 2024

Crossrefs

Cf. A000045, A001065, A005478, A074283, A139589.

Programs

Maple

isA000045 := proc(n) local i,f ; for i from 0 do f := combinat[fibonacci](i) ; if f = n then RETURN(true) ; elif f > n then RETURN(false) ; fi ; od; end: A001065 := proc(n) numtheory[sigma](n)-n ; end: isA139095 := proc(n) RETURN( isA000045(n) and isA000045(A001065(n)) ) ; end: for i from 1 to 230 do if isA139095(combinat[fibonacci](i)) then printf("%d,", combinat[fibonacci](i)) ; fi ; od: # R. J. Mathar, May 22 2008

Mathematica

Fsum[n_]:=DivisorSigma[1,n]-n;Select[Fibonacci[Range[140]],IntegerQ[Sqrt[5*Fsum[#]^2 + 4]] || IntegerQ[Sqrt[5*Fsum[#]^2 - 4]]&] (* James C. McMahon, Jun 28 2025 *)

Extensions

More terms from R. J. Mathar, May 22 2008