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A214434
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Composite numbers k such that k divides Fibonacci(k+1) or Fibonacci(k-1) and 2^(k-1) == 1 (mod k).
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1
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6601, 13981, 30889, 68101, 219781, 252601, 332949, 399001, 512461, 642001, 721801, 722261, 741751, 852841, 873181, 1024651, 1141141, 1193221, 1207361, 1533601, 1690501, 1735841, 1857241, 1909001, 2085301, 2100901, 2165801, 2603381, 2704801, 2757241, 3186821, 3568661
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OFFSET
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1,1
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COMMENTS
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Pseudoprimes to a criterion for primality which tests that
1. k divides Fibonacci(k+1) or Fibonacci(k-1) (see A182554, A081264), and
2. 2^(k-1) == 1 (mod k) (see A001567).
All terms appear to be congruent to 1 or -1 (mod 5).
Terms that are not congruent to 1 or -1 (mod 5): 22711873, 40160737, 55462177, ... . - Amiram Eldar, Sep 12 2022
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..671 from Giovanni Resta)
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FORMULA
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Intersection of A182554 and A001567.
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EXAMPLE
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6601 is in the sequence because the 6600th Fibonacci number is divisible by 6601 and 2^6600 = 1 mod 6601.
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MAPLE
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with(combinat):f:= n-> fibonacci(n): for n from 1 to 2000000 do if(f(n+1) mod n = 0 or f(n-1) mod n = 0) and 2^(n-1) mod n = 1 and not isprime(n) then print(n) fi od;
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MATHEMATICA
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Select[Range[1, 4*10^6, 2], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* Amiram Eldar, Sep 12 2022 *)
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CROSSREFS
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Cf. A182554, A081264, A001567.
Sequence in context: A186563 A252637 A164971 * A317247 A290281 A178213
Adjacent sequences: A214431 A214432 A214433 * A214435 A214436 A214437
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KEYWORD
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nonn
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AUTHOR
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Gary Detlefs, Jul 17 2012
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STATUS
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approved
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