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A266960
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Integer averages of first n Fibonacci numbers (beginning with F(0)).
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0
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0, 1, 2, 6, 13, 356, 3126, 28691, 70268, 271396, 6534495, 64591632, 162057126, 26237436541, 66438353080, 7020479040553, 11201604625686, 296414282891996, 32360305554728271, 339791857819043616, 871053578019254406, 5731478440138170841, 9181907843495831675
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OFFSET
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1,3
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COMMENTS
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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?
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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), ", ")));
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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