login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A339669 Number of Fibonacci divisors of Lucas(n)^2 + 1. 2
2, 2, 3, 1, 3, 2, 3, 2, 5, 1, 5, 2, 4, 2, 5, 1, 5, 2, 4, 2, 6, 1, 6, 2, 4, 2, 6, 1, 6, 2, 4, 2, 6, 1, 7, 2, 5, 2, 6, 1, 6, 2, 4, 2, 7, 1, 7, 2, 5, 2, 7, 1, 6, 2, 5, 2, 7, 1, 6, 2, 4, 2, 8, 1, 9, 2, 5, 2, 6, 1, 6, 2, 4, 2, 7, 1, 9, 2, 6, 2, 7, 1, 7, 2, 5, 2, 7, 1, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156.

Conjecture 1: a(6*n+3) = 1.

Conjecture 2: a(6*n+1) = a(6*n+5) = 2.

The table in the links shows an array where terms are arranged in a table of 12 columns and 13 rows. We see the periods when a(n) = 1 and 2.

LINKS

Table of n, a(n) for n=0..88.

Michel Lagneau, Table

EXAMPLE

a(8) = 5 because the divisors of Lucas(8)^2 + 1 = 47^2 + 1 = 2210 are {1, 2, 5, 10, 13, 17, 26, 34, 65, 85, 130, 170, 221, 442, 1105, 2210} with 5 Fibonacci divisors: 1, 2, 5, 13 and 34.

MAPLE

with(combinat, fibonacci):nn:=100:F:={}:

Lucas:=n->2*fibonacci(n-1)+fibonacci(n):

for k from 0 to nn do:

  F:=F union {fibonacci(k)}:

od:

   for m from 0 to 90 do:

    l:=Lucas(m)^2+1:d:=numtheory[divisors](l):n0:=nops(d):

    lst:= F intersect d: n1:=nops(lst):printf(`%d, `, n1):

   od:

MATHEMATICA

Array[DivisorSum[LucasL[#]^2 + 1, 1 &, AnyTrue[Sqrt[5 #^2 + 4 {-1, 1}], IntegerQ] &] &, 89, 0] (* Michael De Vlieger, Dec 12 2020 *)

PROG

(PARI) a(n) = { my(l2 = 5*fibonacci(n)^2 + 4*(-1)^n + 1, k = 1, m = 2, res = 1, g); while(m <= l2, if(l2 % m == 0, res++); g = m; m += k; k = g; ); res } \\ David A. Corneth, Dec 12 2020

CROSSREFS

Cf. A000032, A000045, A001254, A339461.

Sequence in context: A233431 A160650 A304092 * A171691 A088904 A241568

Adjacent sequences:  A339666 A339667 A339668 * A339670 A339671 A339672

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 12 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)