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A124134
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Positive integers n such that Fibonacci(n) = a^2 + b^2, where a, b are integers.
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3
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1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122, 123, 125, 127
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OFFSET
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1,2
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COMMENTS
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All odd numbers are in this sequence, since the Fibonacci number with index 2m+1 is the sum of the squares of the two Fibonacci numbers with indices m and m+1. Those with even indices ultimately depend on certain Lucas numbers being the sum of two squares (see A124132). Joint work with Kevin O'Bryant and Dennis Eichhorn.
Numbers n such that Fibonacci(n) or Fibonacci(n)/2 is a square are only 0, 1, 2, 3, 6, 12. So a and b must be distinct and nonzero for all values of this sequence except 1, 2, 3, 6, 12. - Altug Alkan, May 04 2016
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LINKS
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FORMULA
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EXAMPLE
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14 is in the sequence because F_14=377=11^2+16^2.
16 is not in the sequence because F_16=987 is congruent to 3 (mod 4).
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MATHEMATICA
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PROG
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(PARI) for(n=1, 10^6, t=fibonacci(n); s=sqrtint(t); forstep(i=s, 1, -1, if(issquare(t-i*i), print1(n, ", "); break))) \\ Ralf Stephan, Sep 15 2013
(PARI) is2s(n)={my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1; } \\ see A001481
for(n=1, 10^6, if(is2s(fibonacci(n)), print1(n, ", "))); \\ Joerg Arndt, Sep 15 2013
(Haskell)
a124134 n = a124134_list !! (n-1)
a124134_list = filter ((> 0) . a000161 . a000045) [1..]
(Python)
from itertools import count, islice
from sympy import factorint, fibonacci
def A124134_gen(): # generator of terms
return filter(lambda n:n & 1 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()), count(1))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006
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EXTENSIONS
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STATUS
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approved
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