A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.

1, 2, 3, 7, 6, 21, 14, 50, 52, 122, 90, 427, 234, 784, 1038, 2351, 1598, 6860, 4182, 17262, 17262, 35622, 28658, 139703, 90031, 243308, 300405, 766850, 514230, 2367006, 1346270, 5188658, 5326470, 11409346, 11782764, 44717548, 24157818

Offset: 1

Author

Leroy Quet, Oct 10 2005

Keywords

• nonn

Comments

a(n) = a(n+1) for n = 20, but for no other n < 25000. - Klaus Brockhaus, Oct 11 2005

If k|n then F(k)|F(n). Therefore A111075(n) = F(n) * sum{k|n} 1/F(k) = sum{k|n} F(n)/F(k) is a sum of integers. - Max Alekseyev, Oct 22 2005

Examples

			a(6) = F(6) sum{k|6} 1/F(k) = F(6) * (1/F(1) + 1/F(2) + 1/F(3) + 1/F(6)) = 8 * (1/1 + 1/1 + 1/2 + 1/8) = 21.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..4785

Crossrefs

Cf. A000045, A007435, A111159, A000204 (Lucas), A203318.

Programs

Maple

with(combinat): with(numtheory): a:=proc(n) local div: div:=divisors(n): fibonacci(n)*sum(1/fibonacci(div[j]),j=1..tau(n)) end: seq(a(n),n=1..40); # Emeric Deutsch, Oct 11 2005
# second Maple program:
a:= n-> (F-> F(n)*add(1/F(d),d=numtheory[divisors(n)))(
         combinat[fibonacci]):
seq(a(n), n=1..42);  # Alois P. Heinz, Aug 20 2019

Mathematica

f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* Robert G. Wilson v, Oct 11 2005 *)

PARI

{for(n=1,37,d=divisors(n);print1(fibonacci(n)*sum(j=1,length(d), 1/fibonacci(d[j])),","))}

PARI

{a(n)=fibonacci(n) * sumdiv(n,d, 1/fibonacci(d))} /* Paul D. Hanna, Oct 11 2005 */

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n, x^m/(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))),n)} /* Paul D. Hanna, Oct 11 2005 */

Formula

G.f.: Sum_{n>=1} x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n). [Paul D. Hanna, Jan 09 2012]

Extensions

More terms from Robert G. Wilson v, Emeric Deutsch, Paul D. Hanna and Klaus Brockhaus, Oct 11 2005