A066769 a(n) = Sum_{d|n} d*Fibonacci(n/d).

1, 3, 5, 9, 10, 21, 20, 39, 49, 80, 100, 195, 246, 424, 650, 1065, 1614, 2715, 4200, 6940, 11020, 17922, 28680, 46821, 75075, 121898, 196565, 318680, 514258, 833560, 1346300, 2180439, 3524900, 5706132, 9227600, 14936241, 24157854, 39096588

Offset: 1

Author

Vladeta Jovovic, Jan 17 2002

Keywords

• easy
• nonn

Comments

Dirichlet convolution of f(n)=n with the Fibonacci numbers F(n)=A000045. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.

Links

• Robert Israel, Table of n, a(n) for n = 1..4740

Crossrefs

Cf. A000045, A007435.

Programs

Maple

N:= 100:
A:= Vector(N):
for k from 1 to N do
  f:= combinat:-fibonacci(k);
  ds:= [$1..floor(N/k)];
  A[k*ds] := A[k*ds] + f*Vector(ds);
od:
convert(A,list); # Robert Israel, Feb 08 2016

Mathematica

a[n_] := DivisorSum[n, # * Fibonacci[n/#] &]; Array[a, 38] (* Amiram Eldar, Sep 16 2020 *)

PARI

a(n) = sumdiv(n, d, d*fibonacci(n/d)); \\ Michel Marcus, Sep 16 2020

Formula

G.f.: Sum_{i>0} i*x^i/(1-x^i-x^(2*i)). - Vladeta Jovovic, Oct 06 2003