A215367 Lengths of binary representations of prime Fibonacci numbers.

2, 2, 3, 4, 7, 8, 11, 15, 19, 29, 32, 57, 90, 94, 249, 299, 300, 311, 353, 394, 396, 2062, 3278, 3739, 6463, 6718, 10018, 17745, 21352, 24991, 26041, 35290, 56815, 72833, 90265, 102810, 139616, 275876, 301148, 409631, 412163, 419815, 646697, 728882, 892522, 1135784, 1251758, 1366768

Offset: 1

Author

Alex Ratushnyak, Aug 08 2012

Keywords

• nonn
• base

Comments

Some of the larger entries may only correspond to probable primes.

As of August 2012, the index of last provable Fibonacci prime is A001605(33)=81839, that is, a(n) corresponds to a probable prime for n>33.

Examples

			Tenth prime Fibonacci number is A005478(10) = 433494437, 29 digits in the binary representation, so a(10)=29.
		

Links

• C. K. Caldwell, The Prime Glossary, Fibonacci prime
• Henri & Renaud Lifchitz, Probable primes

Crossrefs

Cf. A005478, A020909.

Programs

Java

import java.math.BigInteger;
public class A215367 {
  public static void main (String[] args) {
    BigInteger prpr = BigInteger.valueOf(0);
    BigInteger prev = BigInteger.valueOf(1), curr;
    int indices[] = {
    // === insert terms of A001605 here, followed by a comma === //
    -1 };
    int ipos = 1, ind = indices[0];
    for (long k=1; ; ++k) {
      if (k==ind) {
        System.out.printf("%d, ",prev.bitLength());
        ind = indices[ipos++];
        if (ind<0) break;
      }
      curr = prpr.add(prev);
      prpr = prev;
      prev = curr;
    }
  }
}

Mathematica

Length /@ IntegerDigits[Select[Fibonacci[Range[1000]], PrimeQ[#] &], 2] (* T. D. Noe, Aug 08 2012 *)
IntegerLength[#,2]&/@Select[Fibonacci[Range[1000]],PrimeQ] (* Harvey P. Dale, Nov 20 2021 *)

Formula

a(n) = A070939(A005478(n)) = A070939(A000045(A001605(n))).