A266960 Integer averages of first n Fibonacci numbers (beginning with F(0)).

0, 1, 2, 6, 13, 356, 3126, 28691, 70268, 271396, 6534495, 64591632, 162057126, 26237436541, 66438353080, 7020479040553, 11201604625686, 296414282891996, 32360305554728271, 339791857819043616, 871053578019254406, 5731478440138170841, 9181907843495831675

Offset: 1

Author

Altug Alkan, Jan 07 2016

Keywords

• nonn
• easy

Comments

It seems only 0, 1, 2, 13 are Fibonacci numbers.

Are there other Fibonacci numbers of the form (Fibonacci(k) - 1) / (k - 1)?

2 and 13 are the prime numbers. Are there other prime numbers in this sequence?

Examples

			1 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3)) / 4 = 4 / 4 = 1.
2 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3) + Fibonacci(4) + Fibonacci(5)) / 6 = 12 / 6 = 2.
		

Crossrefs

Cf. A101907, A219612.

Programs

Mathematica

Table[Mean@ Fibonacci@ Range[0, n], {n, 0, 100}] /. Rational -> Nothing (* _Michael De Vlieger, Jan 07 2016 *)
Module[{nn=100,fibs},fibs=Accumulate[Fibonacci[Range[0,nn]]];Select[ #[[1]] / #[[2]]&/@Thread[{fibs,Range[nn+1]}],IntegerQ]] (* Harvey P. Dale, Nov 15 2020 *)

PARI

m(n) = sum(k=0, n, fibonacci(k)) % (n+1);
b(n) = sum(k=0, n, fibonacci(k)) / (n+1);
for(n=0, 1e2, if(m(n)==0, print1(b(n), ", ")));

Formula

a(n) = A000071(A219612(n) + 1) / A219612(n).