A339669 -id:A339669 - cp's OEIS frontend

1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 4, 4, 3, 5, 5, 3, 5, 5, 3, 5, 5, 3, 5, 6, 4, 5, 6, 4, 5, 5, 3, 5, 5, 5, 7, 5, 5, 7, 5, 3, 5, 5, 3, 7, 7, 3, 7, 8, 4, 5, 6, 4, 5, 7, 5, 5, 7, 5, 5, 5, 3, 7, 7, 5, 9, 7, 5, 7, 5, 3, 5, 5, 3, 7, 7, 5, 9, 7, 5, 8, 6, 3, 6, 8, 5, 5, 7

Offset: 0

Michel Lagneau, Jan 12 2021

nonn

A Fibonacci divisor of a number k is a Fibonacci number that divides k.
It is interesting to compare this sequence with A339669.
We observe that a(2n) = A339669(2n) if n = 5*k + 2 or n = 5*k + 3, with k >= 0, because Lucas(2n)^2 = 5*Fibonacci(2n)^2 + 4 (see A005248: all nonnegative integer solutions of the Pell equation a(n)^2 - 5*b(n)^2 = +4 together with b(n)= A001906(n), n>=0. - Wolfdieter Lang, Aug 31 2004).
So, Lucas(2n)^2 + 1 = 5*(Fibonacci(2n)^2 + 1). Lucas(2n)^2 + 1 and Fibonacci(2n)^2 + 1 have the same Fibonacci divisors for n = 5*k + 2 or n = 5*k + 3. For the other values of n = 5*k, 5*k + 1 or 5*k + 4, 5 is a Fibonacci divisor of Lucas(2n)^2 + 1 but not of Fibonacci(2n)^2 + 1. So for these last three values of n, a(2n) = A339669(2n) - 1 (except for m = 1 and 2, 5*F(m) is never a Fibonacci number).

			a(13) = 5 because the 5 Fibonacci divisors of Fibonacci(13)^2 + 1 = 233^2 + 1 are 1, 2, 5, 89 and 610.
a(16) = 5 because the 5 Fibonacci divisors of Fibonacci(16)^2 + 1 = 987^2 + 1 are 1, 2, 5, 610, and 1597.
Remark: the 5 Fibonacci divisors of Lucas(16)^2 + 1 = 2207^2 + 1 are 1, 2, 5, 610, and 1597, the index 16 = 2*8 with 8 of the form 5*k + 3.

Michel Marcus, Table of n, a(n) for n = 0..200

Cf. A000032, A000045, A005248, A010056, A339461, A339669.

Cf. A005086, A245306.

with(combinat,fibonacci):nn:=100:F:={}:
for k from 0 to nn do:
  F:=F union {fibonacci(k)}:
od:
   for m from 0 to 90 do:
    f:=fibonacci(m)^2+1:d:=numtheory[divisors](f):
    lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
   od:

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
a(n) = sumdiv(fibonacci(n)^2+1, d, isfib(d)); \\ Michel Marcus, Jan 12 2021

a(n) = A005086(A245306(n)). - Michel Marcus, Aug 10 2022

cp's OEIS Frontend

A340542 Number of Fibonacci divisors of Fibonacci(n)^2 + 1.

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