A133021 -id:A133021 - cp's OEIS frontend

A139044 Smallest prime divisor of the Fibonacci numbers > 1.

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

A139589 Fibonacci numbers with Fibonacci number of divisors.

Original entry on oeis.org

1, 1, 2, 3, 5, 13, 89, 233, 610, 987, 1597, 10946, 28657, 514229, 3524578, 9227465, 24157817, 39088169, 63245986, 433494437, 1836311903, 2971215073, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853

Offset: 1

Omar E. Pol, May 09 2008

nonn

A000005(a(n)) is a Fibonacci number.
For the terms shown here (in the Data section) the number of divisors is 1 or 2 or 8. - Emeric Deutsch, May 12 2008
Up to n = 104 the number of divisors is still 1, 2 or 8. - Amiram Eldar, Oct 15 2019

Amiram Eldar, Table of n, a(n) for n = 1..104

Cf. A000005, A000045, A063375, A133021.

A000045 := proc(n) option remember ; coeftayl( x/(1-x-x^2),x=0,n) ; end: isA000045 := proc(n) local a; for a from 0 do if A000045(a) > n then RETURN(false) ; elif A000045(a)=n then RETURN(true) ; fi ; od: end: A000005 := proc(n) numtheory[tau](n) ; end: isA139589 := proc(n) RETURN(isA000045(n) and isA000045(A000005(n))) ; end: for i from 1 to 130 do a000045 := A000045(i) ; if isA139589(a000045) then printf("%d,",a000045) ; fi ; od: # R. J. Mathar, May 11 2008
with(combinat): with(numtheory): F:={seq(fibonacci(k),k=1..100)}: a:=proc(n) if member(tau(fibonacci(n)),F)=true then fibonacci(n) else end if end proc: seq(a(n),n=1..70); # Emeric Deutsch, May 12 2008

With[{s = Array[Fibonacci, 80]}, Select[s, ! FreeQ[s, DivisorSigma[0, #]] &]] (* Michael De Vlieger, Oct 15 2019 *)

More terms from R. J. Mathar and Emeric Deutsch, May 11 2008

A139045 Largest proper divisor of the Fibonacci numbers > 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 17, 11, 1, 72, 1, 29, 305, 329, 1, 1292, 113, 2255, 5473, 199, 1, 23184, 15005, 521, 98209, 105937, 1, 416020, 2417, 726103, 1762289, 3571, 1845493, 7465176, 330929, 1056437, 31622993, 34111385, 59369, 133957148, 1, 233802911, 567451585

Offset: 3

Omar E. Pol, Apr 23 2008

nonn

See the list of divisors of positive Fibonacci numbers in the triangle A133021.
See the largest proper divisor of n in A032742.
Fibonacci(1)=Fibonacci(2)=1 do not have proper divisors. - Emeric Deutsch, May 18 2008

			a(9) = 17 because the 9th Fibonacci number is 34 and the divisors of 34 are 1, 2, 17, 34, then the largest proper divisor of 34 is 17.

Amiram Eldar, Table of n, a(n) for n = 3..1000

Cf. A000045, A032742, A060383, A133021, A133028, A134708.

with(combinat): with(numtheory): a:=proc(n) options operator, arrow: op(tau(fibonacci(n))-1, divisors(fibonacci(n))) end proc: seq(a(n),n=3..40); # Emeric Deutsch, May 18 2008
# second Maple program:
a:= n-> (f-> f/min(numtheory[factorset](f)))((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=3..47);  # Alois P. Heinz, Sep 03 2019

lpd[n_]:=Divisors[n][[-2]]; lpd/@(Fibonacci[Range[3,40]]) (* Harvey P. Dale, Mar 29 2015 *)

a(n) = A032742(A000045(n)).

a(n) = A000045(n)/A060383(n). - Alois P. Heinz, Sep 03 2019

More terms from Emeric Deutsch, May 18 2008

A139590 Fibonacci numbers with a non-Fibonacci number of divisors.

Original entry on oeis.org

8, 21, 34, 55, 144, 377, 2584, 4181, 6765, 17711, 46368, 75025, 121393, 196418, 317811, 832040, 1346269, 2178309, 5702887, 14930352, 102334155, 165580141, 267914296, 701408733, 1134903170, 4807526976, 12586269025, 32951280099

Offset: 1

Omar E. Pol, May 09 2008

nonn

A000005(a(n)) is a non-Fibonacci number A001690.

			34 belongs to the sequence because the number of its divisors, 4, is not a Fibonacci number.

Cf. A000005, A000045, A001690, A063375, A133021.

A000045 := proc(n) option remember ; coeftayl( x/(1-x-x^2),x=0,n) ; end: isA000045 := proc(n) local a; for a from 0 do if A000045(a) > n then RETURN(false) ; elif A000045(a)=n then RETURN(true) ; fi ; od: end: A000005 := proc(n) numtheory[tau](n) ; end: isA139590 := proc(n) RETURN(isA000045(n) and not isA000045(A000005(n))) ; end: for i from 1 to 130 do a000045 := A000045(i) ; if isA139590(a000045) then printf("%d,",a000045) ; fi ; od: # R. J. Mathar, May 11 2008
with(combinat): with(numtheory): F:={seq(fibonacci(j),j=1..30)}: a:= proc(n) if member(tau(fibonacci(n)),F) = false then fibonacci(n) else end if end proc: seq(a(n),n=1..50); # Emeric Deutsch

With[{fibs=Fibonacci[Range[60]]},Transpose[Select[Thread[{fibs, DivisorSigma[ 0,fibs]}], !MemberQ[ fibs,#[[2]]]&]][[1]]] (* Harvey P. Dale, Aug 04 2013 *)

More terms from R. J. Mathar and Emeric Deutsch, May 11 2008

A238899 Irregular triangle read by rows: row n lists divisors of n-th Lucas number A000032(n).

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 7, 1, 11, 1, 2, 3, 6, 9, 18, 1, 29, 1, 47, 1, 2, 4, 19, 38, 76, 1, 3, 41, 123, 1, 199, 1, 2, 7, 14, 23, 46, 161, 322, 1, 521, 1, 3, 281, 843, 1, 2, 4, 11, 22, 31, 44, 62, 124, 341, 682, 1364, 1, 2207, 1, 3571, 1, 2, 3, 6, 9, 18, 27, 54

Offset: 0

T. D. Noe, Mar 14 2014

Note that, in general, the Lucas numbers have fewer divisors than Fibonacci numbers. Why?

			Triangle begins:
  1,   2;
  1;
  1,   3;
  1,   2,  4;
  1,   7;
  1,  11;
  1,   2,  3,   6,  9, 18;
  1,  29;
  1,  47;
  1,   2,  4,  19, 38, 76;
  1,   3, 41, 123;
  1, 199;
  1,   2,  7,  14, 23, 46, 161, 322;
  ...

T. D. Noe, Rows 0 to 100 of irregular triangle, flattened

Cf. A000032 (Lucas numbers), A027750.
Cf. A133021 (similar triangle for Fibonacci numbers).
Column 2 gives A280104 (for n>=2).

[Divisors(Lucas(n)): n in [0..30]]; // Vincenzo Librandi, Nov 15 2024

Flatten[Table[Divisors[LucasL[n]], {n, 0, 20}]] (* Typo corrected by Harvey P. Dale, Jun 29 2021 *)

A138881 Array read by rows: row n lists divisors of n-th positive triangular number A000217(n).

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 6, 1, 2, 5, 10, 1, 3, 5, 15, 1, 3, 7, 21, 1, 2, 4, 7, 14, 28, 1, 2, 3, 4, 6, 9, 12, 18, 36, 1, 3, 5, 9, 15, 45, 1, 5, 11, 55, 1, 2, 3, 6, 11, 22, 33, 66, 1, 2, 3, 6, 13, 26, 39, 78, 1, 7, 13, 91, 1, 3, 5, 7, 15, 21, 35, 105

Offset: 1

Omar E. Pol, Apr 11 2008

			Array begins:
1
1, 3
1, 2, 3, 6
1, 2, 5, 10
1, 3, 5, 15
1, 3, 7, 21
1, 2, 4, 7, 14, 28

Cf. A000217, A133021, A133031, A138882.

A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

Original entry on oeis.org

1, 2, 4, 1, 2, 3, 6, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

Subsequence of A027750.
Row sums give A073255.
Right border gives A002808.

			Triangle begins:
1,  2,  4;
1,  2,  3,  6;
1,  2,  4,  8;
1,  3,  9;
1,  2,  5,  10;
1,  2,  3,  4,  6,  12;
1,  2,  7,  14;
1,  3,  5,  15
1,  2,  4,  8,  16;
1,  2,  3,  6,  9,  18;
1,  2,  4,  5,  10, 20;
1,  3,  7,  21;
1,  2,  11, 22;
1,  2,  3,  4,  6,  8,  12, 24;
1,  5,  25;
1,  2,  13, 26;
1,  3,  9,  27;
1,  2,  4,  7,  14, 28;
1,  2,  3,  5,  6,  10, 15, 30;
1,  2,  4,  8,  16, 32;
1,  3,  11, 33;
1,  2,  17, 34;
...

Ilya Gutkovskiy, Extended example
Ilya Gutkovskiy, Graphic additions
Eric Weisstein's World of Mathematics, Composite Number
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A002808, A027750, A035004 (row length), A133021, A133031, A138881.

Flatten[Table[Divisors[Composite[n]], {n, 22}]]

tabf(nn) =  forcomposite(c=1, nn, print(divisors(c), ", ")); \\ Michel Marcus, Feb 21 2016

A335534 a(n) = tribonacci(n) modulo Fibonacci(n).

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 0, 3, 10, 26, 60, 130, 38, 173, 485, 175, 977, 273, 2789, 2065, 336, 15149, 22718, 39800, 5226, 54214, 2323, 251416, 418400, 93831, 977776, 1518664, 261912, 5208104, 2557037, 3549042, 21177270, 11203146, 36247269, 87596844, 44950918, 261069681

Offset: 1

Richard Peterson, Jun 12 2020

nonn

a(n) is congruent to tribonacci(n) modulo k if Fibonacci(n) is divisible by k, although the converse does not hold.

			For n=10, since tribonacci(10)=81 and Fibonacci(10)=55, a(10)=81 modulo 55 = 26.

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000073, A000045, A001177, A133021, A046738.

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3] mod (<<0|1>, <1|1>>^n)[1, 2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 19 2020

m = 42; Mod[LinearRecurrence[{1, 1, 1}, {0, 1, 1}, m], Array[Fibonacci, m]] (* Amiram Eldar, Aug 19 2020 *)

t(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = t(n) % fibonacci(n); \\ Michel Marcus, Aug 19 2020

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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A139044 Smallest prime divisor of the Fibonacci numbers > 1.

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A139589 Fibonacci numbers with Fibonacci number of divisors.

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A139045 Largest proper divisor of the Fibonacci numbers > 1.

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A139590 Fibonacci numbers with a non-Fibonacci number of divisors.

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A238899 Irregular triangle read by rows: row n lists divisors of n-th Lucas number A000032(n).

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A138881 Array read by rows: row n lists divisors of n-th positive triangular number A000217(n).

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A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

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A335534 a(n) = tribonacci(n) modulo Fibonacci(n).

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