A034748 Dirichlet convolution of Fibonacci numbers with phi(n).
1, 2, 4, 6, 9, 14, 19, 30, 44, 68, 99, 168, 245, 402, 636, 1026, 1613, 2650, 4199, 6854, 10996, 17820, 28679, 46596, 75065, 121650, 196516, 318250, 514257, 832826, 1346299, 2179374, 3524796, 5704516, 9227571, 14933352, 24157853, 39092386
Offset: 1
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..2000 (first 300 terms from Paolo P. Lava)
Programs
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Magma
[&+[Fibonacci(d)*EulerPhi(n div d): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Aug 19 2018
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Mathematica
Table[Sum[Fibonacci[d] EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 50}] (* Vincenzo Librandi, Aug 19 2018 *) -
PARI
a(n)=sumdiv(n,d,fibonacci(d)*eulerphi(n/d))
Formula
From definition a(n) = Sum_{d|n} F(d)*phi(n/d); also a(n) = Sum_{k=1..n} gcd(F(k), F(k+n)) where F(k) denotes the k-th Fibonacci number. - Benoit Cloitre, May 25 2003
G.f.: Sum_{k>=1} phi(k) * x^k/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Jul 23 2019
a(n) ~ phi^n / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 23 2019
From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{k=1..n} F(gcd(n,k)).
a(n) = Sum_{k=1..n} F(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)