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a(n) is the least m > 0 such that Fibonacci(n+1)-m and n-m are relatively prime.
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%I #9 Feb 17 2018 20:01:28

%S 2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,2,1,1,2,1,2,3,1,1,2,1,2,1,1,1,2,1,2,1,

%T 1,1,2,1,2,2,2,1,2,1,2,3,1,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,2,1,1,2,1,

%U 2,2,1,1,2,1,2,1,1,2,2,1,2,1,1,1,2,1,2

%N a(n) is the least m > 0 such that Fibonacci(n+1)-m and n-m are relatively prime.

%H Clark Kimberling, <a href="/A214713/b214713.txt">Table of n, a(n) for n = 1..1000</a>

%e gcd(21-1,7-1) = 2 and gcd(21-2,7-2) = 1, so a(7) = 2.

%t Table[m = 1; While[GCD[Fibonacci[n+1] - m, n - m] != 1, m++]; m, {n, 1, 140}]

%Y Cf. A214712.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jul 27 2012