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Smallest k such that n/k is a Fibonacci number.
5

%I #15 May 06 2024 11:04:26

%S 1,1,1,2,1,2,7,1,3,2,11,4,1,7,3,2,17,6,19,4,1,11,23,3,5,2,9,14,29,6,

%T 31,4,11,1,7,12,37,19,3,5,41,2,43,22,9,23,47,6,49,10,17,4,53,18,1,7,

%U 19,29,59,12,61,31,3,8,5,22,67,2,23,14,71,9,73,37,15,38,77,6,79,10,27,41

%N Smallest k such that n/k is a Fibonacci number.

%H Charles R Greathouse IV, <a href="/A054495/b054495.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n/A054494(n). [Corrected by _Charles R Greathouse IV_, Nov 05 2014]

%e a(10)=2 because 10/1=10 is not a Fibonacci number but 10/2=5 is.

%o (PARI) A010056(n)=my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))

%o a(n)=fordiv(n,d,if(A010056(n/d), return(d))) \\ _Charles R Greathouse IV_, Nov 05 2014

%o (Python)

%o from sympy import divisors

%o from sympy.ntheory.primetest import is_square

%o def A054495(n): return next(d for d in divisors(n) if is_square(m:=5*(n//d)**2-4) or is_square(m+8)) # _Chai Wah Wu_, May 06 2024

%Y Cf. A000045, A010056, A005086, A037943.

%K easy,nonn

%O 1,4

%A _Henry Bottomley_, Apr 04 2000

%E a(34), a(55), a(68) corrected by _Charles R Greathouse IV_, Nov 06 2014