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

0,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A300778 A199084 A344371 * A240214 A181132 A278168

Adjacent sequences: A322001 A322002 A322003 * A322005 A322006 A322007

Keyword

nonn

Author

M. F. Hasler, Dec 12 2018

Status

approved