A139045 -id:A139045 - cp's OEIS frontend

A105800 Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 8, 1, 13, 5, 21, 1, 34, 1, 55, 13, 89, 1, 144, 5, 233, 34, 377, 1, 610, 1, 987, 89, 1597, 13, 2584, 1, 4181, 233, 6765, 1, 10946, 1, 17711, 610, 28657, 1, 46368, 13, 75025, 1597, 121393, 1, 196418, 89, 317811, 4181, 514229, 1, 832040, 1

Offset: 1

Reinhard Zumkeller, Apr 20 2005

nonn

a(A001605(n)) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..987
Eric Weisstein's World of Mathematics, Fibonacci Number
Wikipedia, Divisibility sequence

Cf. A000045, A032742, A076984, A139045, A280686, A280688, A280689, A280690, A280698, A280699.

Cf. A046022 (gives the positions of ones).

nn=70;Join[{1,1},With[{fibs=Fibonacci[Range[nn]]},Table[ Max[ Intersection[ Most[Divisors[fibs[[n]]]],fibs]],{n,3,nn}]]] (* Harvey P. Dale, Apr 10 2012 *)

From Antti Karttunen, Jan 11 2017: (Start)
a(n) = A280686(A000045(n)).
a(n) = A000045(A032742(n)). [Because A000045 is a divisibility sequence.]
a(A032742(n)) = A280688(n).
(End)

A139044 Smallest prime divisor of the Fibonacci numbers > 1.

Original entry on oeis.org

2, 3, 5, 2, 13, 3, 2, 5, 89, 2, 233, 13, 2, 3, 1597, 2, 37, 3, 2, 89, 28657, 2, 5, 233, 2, 3, 514229, 2, 557, 3, 2, 1597, 5, 2, 73, 37, 2, 3, 2789, 2, 433494437, 3, 2, 139, 2971215073, 2, 13, 5, 2, 3, 953, 2, 5, 3, 2, 59, 353, 2, 4513, 557, 2, 3, 5, 2, 269, 3, 2, 5, 6673, 2

Offset: 1

Omar E. Pol, Apr 23 2008

nonn

Fibonacci number > 1, divided by its largest proper divisor.

Tyler Busby, Table of n, a(n) for n = 1..1450 (terms 1..200 Vincenzo Librandi, terms 201..1406 from Amiram Eldar)

Cf. A000045, A032742, A060383, A133021, A139045.

[Minimum(PrimeDivisors(Fibonacci(n+2))): n in [1..70]]; // Vincenzo Librandi, Dec 24 2016

with(numtheory): with(combinat): a:=proc(n) options operator, arrow: op(2, divisors(fibonacci(n))) end proc: seq(a(n),n=3..60); # Emeric Deutsch, May 02 2008

First[First[FactorInteger[ # ]]]&/@Fibonacci[Range[3,40]] (* Harvey P. Dale, Apr 30 2008 *)

a(n) = factor(fibonacci(n+2))[1,1]; \\ Michel Marcus, Nov 15 2014

a(n) = A000045(n+2)/A032742(A000045(n+2)) = A000045(n+2)/A139045(n).
a(n) = A020639(A000045(n+2)). - Michel Marcus, Nov 15 2014
a(n) = A060383(n+2). - Alois P. Heinz, Oct 11 2015

More terms from Emeric Deutsch and Harvey P. Dale, May 02 2008

More terms from Vincenzo Librandi, Dec 24 2016

A386994 Number of 2-dense sublists of divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 4, 2, 4, 2, 1, 2, 4, 4, 8, 2, 3, 4, 8, 4, 4, 2, 1, 6, 4, 4, 12, 2, 1, 4, 16, 4, 4, 8, 1, 8, 8, 4, 3, 4, 1, 2, 11, 6, 8, 2, 1, 8, 10, 4, 12, 4, 3, 13, 5, 10, 8, 4, 1, 4, 8, 10, 17, 8, 7, 8, 20, 9, 15, 4, 1, 4, 16, 18, 24, 15, 7, 4, 3, 5

Offset: 0

Omar E. Pol, Aug 27 2025

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

			For n = 18 the 18th Fibonacci number is 2584. The list of divisors of 2584 is [1, 2, 4, 8, 17, 19, 34, 38, 68, 76, 136, 152, 323, 646, 1292, 2584]. There are three 2-dense sublists of divisors of 2584, they are [1, 2, 4, 8], [17, 19, 34, 38, 68, 76, 136, 152] and [323, 646, 1292, 2584], so a(18) = 3.

Cf. A000045, A133021, A139045, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030, A386993.

A386994[n_] := Length[Split[Divisors[Fibonacci[n]], #2 <= 2*# &]];
Array[A386994, 100, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = A237271(A000045(n)), n >= 1. (conjectured).

More terms from Alois P. Heinz, Aug 27 2025

A139227 Array read by rows: row n lists the proper divisors of n-th Fibonacci number A000045(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 1, 3, 7, 1, 2, 17, 1, 5, 11, 1, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 1, 1, 13, 29, 1, 2, 5, 10, 61, 122, 305, 1, 3, 7, 21, 47, 141, 329, 1, 1, 2, 4, 8, 17, 19, 34, 38, 68, 76, 136, 152, 323, 646, 1292

Offset: 3

Omar E. Pol, Apr 28 2008

			Row ....... Array begins
===========================================================
3 ......... 1
4 ......... 1
5 ......... 1
6 ......... 1, 2, 4
7 ......... 1
8 ......... 1, 3, 7
9 ......... 1, 2, 17
10 ........ 1, 5, 11
11 ........ 1
12 ........ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72
13 ........ 1

Cf. A000045, A133021, A139044, A139045.

f[n_] := Most@ Divisors@ Fibonacci@ n; Flatten@ Array[f, 16, 3] (* Robert G. Wilson v *)

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A105800 Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.

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A139044 Smallest prime divisor of the Fibonacci numbers > 1.

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A386994 Number of 2-dense sublists of divisors of the n-th Fibonacci number.

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A139227 Array read by rows: row n lists the proper divisors of n-th Fibonacci number A000045(n).

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Mathematica