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a(n) = gcd(n, A113222(n)), where A113222(n) = Sum_{p^e || n} F(p^e), each p^e the highest power of prime p dividing n (with e > 0), and F(k) is the k-th Fibonacci number.
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%I #12 Jan 14 2025 16:57:57

%S 1,1,1,1,5,3,1,1,1,2,1,1,1,14,1,1,1,1,1,4,3,2,1,1,25,26,1,4,1,2,1,1,1,

%T 34,1,1,1,2,1,2,1,2,1,4,3,46,1,1,1,2,3,4,1,3,1,2,1,2,1,10,1,2,1,1,1,2,

%U 1,4,3,1,1,1,1,74,3,4,1,2,1,16,1,2,1,6,1,86,1,22,1,10,1,4,3,94,1,1,1,2,3

%N a(n) = gcd(n, A113222(n)), where A113222(n) = Sum_{p^e || n} F(p^e), each p^e the highest power of prime p dividing n (with e > 0), and F(k) is the k-th Fibonacci number.

%H Antti Karttunen, <a href="/A111142/b111142.txt">Table of n, a(n) for n = 1..20000</a>

%e 12 = 2^2 * 3^1. So a(12) = GCD(12, F(2^2) + F(3^1)) = GCD(12, 5) = 1.

%t f[n_] := GCD[n, Plus @@ (Fibonacci[ #[[1]]^#[[2]]] & /@ FactorInteger[n])]; Table[ f[n], {n, 99}] (* _Robert G. Wilson v_ *)

%o (PARI)

%o A113222(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1,#f~, fibonacci(f[i, 1]^f[i, 2])));

%o A111142(n) = gcd(n, A113222(n)); \\ _Antti Karttunen_, Jan 14 2025

%Y Cf. A000045, A113222.

%K nonn

%O 1,5

%A _Leroy Quet_, Oct 18 2005

%E More terms from _Robert G. Wilson v_, Oct 21 2005