A322004 Index i of the smallest Fibonacci number > n such that Fib(i) - n is a prime, or 0 if no such index exists.

3, 4, 5, 5, 8, 6, 6, 12, 7, 15, 7, 7, 10, 12, 8, 9, 8, 9, 8, 8, 16, 9, 11, 9, 10, 102, 10, 9, 11, 9, 11, 9, 9, 15, 13, 12, 10, 12, 10, 15, 13, 12, 10, 12, 10, 18, 11, 12, 10, 36, 10, 66, 10, 10, 13, 12, 20, 21, 11, 24, 11, 12, 20, 15, 14, 12, 11, 24, 16, 15, 11, 12, 11, 12, 17, 33, 11, 12, 11, 21, 16, 18, 11, 12, 11, 12, 11, 11, 19, 15, 19, 12, 20, 21, 13, 24

Offset: 0

Author

M. F. Hasler, Dec 12 2018

Keywords

• nonn

Comments

Motivated by a recent "conjecture" seen on internet, that any number > 1 is of the form prime(i) + Fib(j) + Fib(k). Actually the number of such representations increases that fast that this conjecture seems not interesting. The present sequence shows that any number is the difference of a Fibonacci number and a prime, or such that n + some prime = some Fibonacci number. (See A322005 for the corresponding prime.) Most of the terms correspond to the index of the smallest Fibonacci number > n or the subsequent one. Local maxima and/or a(n) > a(n+1) + 1 correspond to numbers for which one has to look further. a(25) = 102 is a noteworthy example.

Examples

			For n = 0, Fibonacci(3) = 2 is the smallest Fibonacci number F such that F - n is a prime, so a(0) = 3.
For n = 1, Fibonacci(4) = 3 is the smallest Fibonacci number F such that F - n = 3 - 1 = 2 is a prime, so a(1) = 4.
For n = 2, Fibonacci(5) = 5 is the smallest Fibonacci number F such that F - n = 5 - 2 = 3 is a prime, so a(2) = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000040 (primes), A000045 (Fibonacci numbers).

Programs

Maple

f:= proc(n) local p,k,a,b,c;
a:= -n:b:= 1-n:
for k from 2 do
  c:= b;
  b:= a+b+n;
  a:= c;
  if isprime(b) then return k fi
od
end proc:
map(f, [$0..100]); # Robert Israel, Dec 14 2018

Mathematica

primeQ[n_] := n>0 && PrimeQ[n]; a[n_] := Module[{i=2}, While[!primeQ[Fibonacci[i]-n], i++]; i]; Array[a,100,0] (* Amiram Eldar, Dec 12 2018 *)

PARI

a(n)=for(i=1,oo,ispseudoprime(fibonacci(i)-n)&&return(i))