A111142 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.

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, 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, 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

Offset

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Example

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

Mathematica

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

Prog

(PARI)
A113222(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, fibonacci(f[i, 1]^f[i, 2])));
A111142(n) = gcd(n, A113222(n)); \\ Antti Karttunen, Jan 14 2025

Crossrefs

Cf. A000045, A113222.

Sequence in context: A265606 A368602 A132199 * A179626 A174965 A159671

Adjacent sequences: A111139 A111140 A111141 * A111143 A111144 A111145

Keyword

nonn

Author

Leroy Quet, Oct 18 2005

Extensions

More terms from Robert G. Wilson v, Oct 21 2005

Status

approved