%I #23 Sep 07 2024 21:10:17
%S 1,1,2,3,5,2,13,21,34,5,89,6,233,13,10,987,1597,34,4181,15,26,89,
%T 28657,42,75025,233,196418,39,514229,10,1346269,2178309,178,1597,65,
%U 102,24157817,4181,466,105,165580141,26,433494437,267,170,28657,2971215073
%N 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.
%C F(p^j) is always coprime to F(q^k), where p and q are distinct primes and j and k are nonnegative integers.
%F Multiplicative with a(p^e) = F(p^e). - _Franklin T. Adams-Watters_, Jun 05 2006
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/F(p^k)) = 4.7856032241... . - _Amiram Eldar_, Jan 20 2024
%e 45 = 3^2 * 5^1, so a(45) = F(3^2) * F(5^1) = 34 * 5 = 170.
%t 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 *)
%o (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
%Y Cf. A000045, A113196.
%K nonn,mult
%O 1,3
%A _Leroy Quet_, Oct 17 2005
%E More terms from Esa Peuha (esa.peuha(AT)helsinki.fi) and Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 26 2005