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
Examples
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.
Links
- Daniel Suteu, Table of n, a(n) for n = 1..88
- Charles Marsden, Python program
Programs
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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 *) -
Python
# See links.
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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
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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
Formula
Extensions
a(36)-a(42) from Daniel Suteu, Nov 02 2024
Comments