A377270 Smallest index k such that the k-th prime number in base-2 contains the n-th Fibonacci number in base-2 as a contiguous substring.

1, 1, 1, 2, 3, 7, 6, 14, 33, 48, 24, 106, 51, 240, 362, 305, 251, 1269, 1047, 1752, 2456, 3773, 3121, 8959, 39089, 62223, 33299, 177305, 42613, 238782, 373418, 699763, 916051, 2715933, 2256419, 13103923, 7100513, 16902825, 13833549, 11323041, 66402079, 54299882

Offset

1,4

Comments

The intersections between this sequence and similar sequences in base-B occur at values of n that are the indices of prime Fibonacci numbers, and values of a(n) such that the a(n)-th prime number is a Fibonacci number.

Links

Daniel Suteu, Table of n, a(n) for n = 1..88

Charles Marsden, Python program

Formula

a(n) = A377483(A000045(n)). - Pontus von Brömssen, Nov 29 2024

Example

For n=1, fib(1)=1 -> 1 in base-2. The first prime containing 1 in its base-2 form is P(1)=2 -> 10. Therefore, a(1)=1.
For n=4, fib(4)=3 -> 11 in base-2. The first prime containing 11 in its base-2 form is P(2)=3 -> 11. Therefore, a(4)=2.
For n=6, fib(6)=8 -> 1000 in base-2. The first prime containing 1000 in its base-2 form is P(7)=17 -> 10001. Therefore, a(6)=7.

Mathematica

s={}; Do[k=0; Until[SequenceCount[IntegerDigits[Prime[k], 2], IntegerDigits[Fibonacci[n], 2]]>0, k++]; AppendTo[s, k], {n, 31}]; s (* James C. McMahon, Nov 21 2024 *)

Prog

(Python) # See links.
(Python)
from sympy import fibonacci, nextprime, primepi
def A377270(n):
    f = fibonacci(n)
    p, k, a = nextprime(f-1), primepi(f-1)+1, bin(f)[2:]
    while True:
        if a in bin(p)[2:]:
            return k
        p = nextprime(p)
        k += 1 # Chai Wah Wu, Nov 20 2024
(Sidef)
for n in (1..35) {
    var bin = n.fib.as_bin
    var min = Inf
    for k in (0..Inf) {
        ['0', '1'].variations_with_repetition(k, {|*a|
            [a..., bin].variations(a.len+1, {|*t|
                var m = Num(t.join, 2)
                if (m.is_prime && m.as_bin.contains(bin)) {
                    min = m if (m < min)
                }
            })
        })
        break if (min < Inf)
    }
    print(min.primepi, ", ")
} # Daniel Suteu, Nov 02 2024

Crossrefs

Cf. A000040, A000045, A001605, A004676, A099000, A377483.

Sequence in context: A019585 A350408 A187015 * A245467 A070964 A111075

Adjacent sequences: A377267 A377268 A377269 * A377271 A377272 A377273

Keyword

nonn,base

Author

Charles Marsden, Oct 22 2024

Extensions

a(36)-a(42) from Daniel Suteu, Nov 02 2024

Status

approved