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

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

Author

Paolo P. Lava, Jan 26 2015

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

Programs

Maple

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);