A221018 Number k such that Fibonacci(k) is divisible by k, k + 1 and k - 1.

108, 2688, 3528, 4968, 6480, 15288, 18048, 20808, 21000, 25308, 35448, 47520, 62928, 68208, 81648, 82008, 97608, 103968, 108288, 116928, 137088, 139968, 151848, 162288, 196560, 197568, 200928, 235008, 238728, 253368, 278208, 363888, 379008, 400248, 444528, 449568, 457608

Offset

1,1

Comments

Numbers k such that Fibonacci(k) mod k = Fibonacci(k) mod (k-1) = Fibonacci(k) mod (k+1) = 0.

Subsequence of A023172.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..200

Prog

(Python)
prpr, prev = 0, 1
for n in range(2, 100000):
    prpr, prev = prev, prpr+prev
    if n%2: continue
    if (prev % n, prev % (n-1), prev % (n+1)) == (0, 0, 0):
        print(n, end=", ")
(Python)
from itertools import count, islice
def A221018_gen(startvalue=2): # generator of terms >= startvalue
    def fibonacci_mod(n, m): # fibonacci(n) mod m
        a, b, c, d, a2, b2, c2, d2 = 1, 1, 1, 0, 1, 0, 0, 1
        for x in bin(n)[2:]:
            e, f = b2*c2%m, a2+d2
            a2, b2, c2, d2 = (pow(a2, 2, m)+e)%m, b2*f%m, c2*f%m, (pow(d2, 2, m)+e)%m
            if x=='1':
                a2, b2, c2, d2 = (a2*a+b2*c)%m, (a2*b+b2*d)%m, (c2*a+d2*c)%m, (c2*b+d2*d)%m
        return c2
    return filter(lambda m: not (fibonacci_mod(m, m-1) or fibonacci_mod(m, m) or fibonacci_mod(m, m+1)), count(max(startvalue, 2)))
A221018_list = list(islice(A221018_gen(), 20)) # Chai Wah Wu, Jul 03 2026

Crossrefs

Cf. A000045, A023172, A225219.

Sequence in context: A063809 A183357 A264681 * A338381 A269182 A244877

Adjacent sequences: A221015 A221016 A221017 * A221019 A221020 A221021

Keyword

nonn,changed

Author

Alex Ratushnyak, May 03 2013

Status

approved