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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 12 2020
STATUS
approved