A219612 -id:A219612 - cp's OEIS frontend

A101907 Numbers k such that the arithmetic mean of the first k+1 Fibonacci numbers (beginning with F(0)) is an integer.

0, 3, 5, 8, 10, 18, 23, 28, 30, 33, 40, 45, 47, 58, 60, 70, 71, 78, 88, 93, 95, 99, 100, 105, 108, 119, 128, 130, 138, 143, 148, 150, 165, 178, 180, 190, 191, 198, 200, 210, 213, 215, 219, 225, 228, 238, 239, 240, 248, 250, 268, 270, 273, 280, 287, 310, 320, 330

Offset: 1

Ctibor O. Zizka, Jul 27 2008

The sum of the first n Fibonacci numbers is F(n+2)-1, sequence A000071.

Knott discusses the factorization of these numbers. - T. D. Noe, Oct 10 2005

			k = 3 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1. So 3 is a term.
k = 5 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2. So 5 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., Ser. A, Vol. 81, No. 2 (2005), pp. 17-20.
H. R. Morton, Fibonacci-like sequences and greatest common divisors, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734.
Morgan Ward, The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386.
Eric W. Weisstein's World of Mathematics, Arithmetic mean.
Eric W. Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A000071. See A111035 for another version.

Cf. A219612. - Altug Alkan, Dec 29 2015

Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* Robert G. Wilson v *)

is(n)=((Mod([1,1;1,0],n+1))^(n+2))[1,2]==1 \\ Charles R Greathouse IV, Feb 04 2013

Numbers k such that (F(0)+ F(1)+ ... + F(k)) / (k+1) is an integer, where F(i) is the i-th Fibonacci number.

a(n) = A219612(n) - 1. - Altug Alkan, Dec 29 2015

Edited and extended by Robert G. Wilson v, Aug 03 2008

Definition corrected by Altug Alkan, Dec 29 2015

A254141 The average of a(n) consecutive Fibonacci numbers is never an integer.

Original entry on oeis.org

8, 16, 21, 28, 32, 40, 52, 55, 56, 64, 65, 68, 69, 80, 84, 85, 87, 88, 92, 93, 99, 104, 105, 112, 117, 119, 128, 132, 133, 136, 140, 141, 145, 148, 152, 153, 155, 156, 160, 161, 164, 165, 171, 172, 176, 184, 187, 188, 196, 200, 203, 204, 205, 207, 208, 209, 212

Offset: 1

Paolo P. Lava, Jan 26 2015

nonn

Subset of A033949 and A175594 (essentially the same sequence).
Numbers of the form 2^k, with k>=3, appear to be part of the sequence.
The file "List of indexes and steps (k, x, y)" (see Links) for k = 1, 2, 3, 4, ... consecutive Fibonacci numbers gives the minimum index to start to calculate the average ( x ) and the step to add to get all the other averages ( y ).
E.g.: for k = 7 we have 7, 6, 8. This means that we must start from the 6th Fibonacci number to add 7 consecutive Fibonacci numbers and get an average that is an integer. Fibonacci(6) + Fibonacci(7) + ... + Fibonacci(12) = 8 + 13 + 21 + 34 + 55 + 89 + 144 = 364 and 364 / 7 = 52.
Then 6 + 1*8 = 14, 6 + 2*8 = 22, 6 + 3*8 = 30, etc. are the other indexes:
Fibonacci(14) + Fibonacci (15) + ... + Fibonacci(20) = 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 = 17101 and 17101 / 7 = 2443;
Fibonacci(22) + Fibonacci(23) + ... + Fibonacci(28) = 17711 + 28657 + 46368 + 75025 + 121393 + 196418 + 317811 = 803383 and 803383 / 7 = 114769;
Fibonacci(30) + Fibonacci(31) + ... + Fibonacci(36) = 832040 + 1346269 + 2178309 + 3524578 + 5702887 + 9227465 + 14930352 = 37741900 and 37741900 / 7 = 5391700; etc.
In particular we note that:
x = 0 is A219612; x = 1 is A124456; x = 0 and y = k - 1 is A106535;
y = 1 is A141767; x = k - 1 and y = k + 1 is A000057;
x = y - 1 or y|k is A023172; y = k is A000351;
x = y - k + 1 appears to give only prime numbers: 3,11,19,31,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971, etc.

Paolo P. Lava, Table of n, a(n) for n = 1..200
Paolo P. Lava, List of indexes and steps (k, x, y)

Cf. A000045, A000057, A000071, A023172, A033949, A106535, A124456, A141767, A175594, A219612.

with(numtheory); with(combinat):P:=proc(q) local a,b,k,j,n,ok;
for j from 1 to q do b:=0; ok:=1;
for n from 0 to q do a:=add(fibonacci(n+k),k=0..j-1)/j;
if type(a,integer) then ok:=0; break; fi; od;
if ok=1 then print(j); fi; od; end: P(20000);

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

Original entry on oeis.org

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

Altug Alkan, Jan 07 2016

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?

			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.

Cf. A101907, A219612.

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 *)

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), ", ")));

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

cp's OEIS Frontend

A101907 Numbers k such that the arithmetic mean of the first k+1 Fibonacci numbers (beginning with F(0)) is an integer.

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A254141 The average of a(n) consecutive Fibonacci numbers is never an integer.

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

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