A113195 a(n) = Product_{p primes} F(p^(m_{n,p})), where p^(m_{n,p}) is highest power of p dividing n, m = nonnegative integer and F(k) is the k-th Fibonacci number.

1, 1, 2, 3, 5, 2, 13, 21, 34, 5, 89, 6, 233, 13, 10, 987, 1597, 34, 4181, 15, 26, 89, 28657, 42, 75025, 233, 196418, 39, 514229, 10, 1346269, 2178309, 178, 1597, 65, 102, 24157817, 4181, 466, 105, 165580141, 26, 433494437, 267, 170, 28657, 2971215073

Offset

1,3

Comments

F(p^j) is always coprime to F(q^k), where p and q are distinct primes and j and k are nonnegative integers.

Links

Robert Israel, Table of n, a(n) for n = 1..4782

Formula

Multiplicative with a(p^e) = F(p^e). - Franklin T. Adams-Watters, Jun 05 2006

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/F(p^k)) = 4.7856032241... . - Amiram Eldar, Jan 20 2024

Example

45 = 3^2 * 5^1, so a(45) = F(3^2) * F(5^1) = 34 * 5 = 170.

Maple

f:= proc(n) local t;
  mul(combinat:-fibonacci(t[1]^t[2]), t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 20 2025

Mathematica

b[t_]:=Fibonacci[First[t]^Last[t]]; a[n_]:=Apply[Times, Map[b, FactorInteger[n]]] (* Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 26 2005 *)

Prog

(PARI) for(n=1, 100, f=factor(n); p=1; for(i=1, matsize(f)[1], p*=fibonacci(f[i, 1]^f[i, 2])); print1(p, ", ")) \\ Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 26 2005

Crossrefs

Cf. A000045, A113196.

Sequence in context: A060442 A060385 A080648 * A069110 A238684 A352377

Adjacent sequences: A113192 A113193 A113194 * A113196 A113197 A113198

Keyword

nonn,mult,look

Author

Leroy Quet, Oct 17 2005

Extensions

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi) and Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 26 2005

Status

approved