A374026 a(n) = the smallest k such that Fibonacci(k) begins and ends with n, where Fibonacci(k) > n, or -1 if there are none.

22, 114, 124, 72, 10, 39, 116, 207, 169, 5715, 2428, 5634, 2366, 189, 2620, 4668, 3137, 2673, 5812, 12090, 721, 11583, 20086, 3798, 1975, 999, 10636, 846, 2071, 9105, 1162, 2076, 8287, 11091, 2770, 2928, 12943, 8493, 172, 2220, 5359, 4737, 28126, 11838, 10460, 7479, 10996

Offset: 1

Author

Gonzalo Martínez, Jun 26 2024

Keywords

• nonn
• base

Comments

Since the smallest number beginning and ending with n is the same n, the condition that Fibonacci(k) > n is imposed. Partial overlap of the start and end is allowed, but not full overlap.

Examples

			As Fibonacci(22) = 17711 is the smallest Fibonacci number greater than 1 that begins and ends with 1, a(1) = 22.
As Fibonacci(10) = 55 is the smallest Fibonacci number greater than 5 that begins and ends with 5, a(5) = 10.
		

Crossrefs

Cf. A000045, A272623, A023183, A020344.

Programs

PARI

isok(s,t) = my(ss=strsplit(s, t)); (#ss >= 3) && (ss[1]=="") && (ss[#ss]=="");
a(n) = my(k=7); while(!isok(Str(fibonacci(k)), Str(n)), k++); k; \\ Michel Marcus, Jun 26 2024

Python

# uses A023183() in A023183
from itertools import count
def a(n):
    if A023183(n) == -1:
        return -1
    f, g, s = 1, 2, str(n)
    pow10 = 10**len(s)
    for k in count(3):
        if g%pow10 == n:
            sfib = str(g)
            if g > n and sfib.startswith(s):
                return k
        f, g = g, f+g
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jul 03 2024

Formula

a(n) >= max{A023183(n), A020344(n)} except that a(n) = -1 when A023183(n) = -1. - Michael S. Branicky, Jun 27 2024

Extensions

More terms from Michel Marcus, Jun 26 2024