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

0, 1, 2, 3, 5, 3, 13, 21, 34, 6, 89, 5, 233, 14, 7, 987, 1597, 35, 4181, 8, 15, 90, 28657, 23, 75025, 234, 196418, 16, 514229, 8, 1346269, 2178309, 91, 1598, 18, 37, 24157817, 4182, 235, 26, 165580141, 16, 433494437, 92, 39, 28658, 2971215073, 989, 7778742049, 75026, 1599, 236, 53316291173, 196419, 94, 34, 4183

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..4782

Formula

Additive with a(p^e) = A000045(p^e).

Example

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

Maple

a:= n-> add(combinat[fibonacci](i[1]^i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 14 2025

Mathematica

f[n_] := Plus @@ (Fibonacci[ #[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[ f[n], {n, 49}] (* 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]))); \\ Antti Karttunen, Jan 14 2025

Crossrefs

Cf. A000045, A113177, A113195, A111142 [= gcd(n, a(n))].

Sequence in context: A272202 A244609 A209195 * A366671 A060444 A002587

Adjacent sequences: A113219 A113220 A113221 * A113223 A113224 A113225

Keyword

nonn

Author

Leroy Quet, Oct 18 2005

Extensions

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

Terms a(49..57) added, and the notation used in the definition simplified by Antti Karttunen, Jan 14 2025

Status

approved