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A182504
Composite numbers k that divide both Fibonacci(k+1) and Fibonacci(2k+1)-1.
1
323, 377, 3827, 5777, 6479, 10877, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 75077, 78089, 79547, 80189, 81719, 82983, 84279, 84419, 86063, 90287, 94667, 100127, 104663, 109871
OFFSET
1,1
COMMENTS
A subset of A182554 based on a refinement of the Fibonacci criterion for primality described there. The additional constraint that k divides Fibonacci(2*k+1)-1 is suggested by the Cloitre comment in A003631.
What base-2 pseudoprimes are contained in this sequence?
An almost identical sequence can be obtained by testing for composite numbers for which (1) k divides Fibonacci(k+1) and (2) k^12 mod 210 = 1. All primes greater than 7 appear to satisfy condition 2. Terms of {a(n)} which are not pseudoprimes to this criterion are 50183, 65471, 82983, and 84279. - Gary Detlefs, Jun 04 2012
LINKS
MAPLE
with (combinat): f:= n-> fibonacci(n): for n from 2 to 100000 do if not isprime(n) and irem(f(n+1), n)=0 and irem((f(2*n+1)-1), n)=0 then print(n) fi od;
MATHEMATICA
Select[Range[110000], CompositeQ[#]&&Mod[Fibonacci[#+1], #]==Mod[Fibonacci[ 2#+1]-1, #] == 0&] (* Harvey P. Dale, Aug 02 2024 *)
PROG
(Magma) [n: n in [4..11*10^4] | not IsPrime(n) and IsDivisibleBy(Fibonacci(n+1), n) and IsDivisibleBy(Fibonacci(2*n+1)-1, n)]; // Bruno Berselli, May 04 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary Detlefs, May 05 2012
STATUS
approved