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A363533
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Least k such that n*F(k)+1 is prime, where F = A000045 is the Fibonacci sequence, or -1 if no such k exists.
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5
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1, 1, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 9, 3, 3, 1, 6, 1, 9, 3, 3, 1, 3, 4, 18, 3, 9, 1, 3, 1, 15, 4, 3, 4, 3, 1, 9, 5, 3, 1, 3, 1, 48, 3, 9, 1, 24, 3, 9, 3, 3, 1, 3, 3, 9, 3, 6, 1, 24, 1, 36, 5, 3, 4, 3, 1, 12, 3, 3, 1, 6, 1, 12, 3, 3, 4, 6, 1, 9, 4, 3, 1, 3, 5
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OFFSET
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1,3
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COMMENTS
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2 does not appear because F(1) = F(2).
a(n) is divisible by 3 if n >= 3 is odd (unless a(n) = -1), because F(k) is odd (so n*F(k)+1 > 2 is even) when k is not divisible by 3.
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LINKS
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FORMULA
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a(n) = 1 if and only if n+1 is prime.
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EXAMPLE
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For n = 17, the least k such that 17*F(k)+1 is prime is k = 6, with 17*F(6)+1 = 17*8+1 = 137, so a(17) = 6.
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MATHEMATICA
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Array[(k = 1; While[! PrimeQ[# Fibonacci[k] + 1], k++]; k) &, 85] (* Michael De Vlieger, Jun 10 2023 *)
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PROG
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(Python)
from sympy import isprime, fibonacci
from itertools import count
# Note: the function hangs if a(n) = -1.
return next(k for k in count(1) if isprime(n*fibonacci(k)+1))
(PARI) a(n) = my(k=1); while(!isprime(n*fibonacci(k)+1), k++); k; \\ Michel Marcus, Jun 10 2023
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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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