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A322004
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Index i of the smallest Fibonacci number > n such that Fib(i) - n is a prime, or 0 if no such index exists.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=for(i=1, oo, ispseudoprime(fibonacci(i)-n)&&return(i))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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