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A242074 Numbers n such that n^2 - 1 is the product of four distinct Fibonacci numbers greater than 1. 1
25, 41, 64, 103, 131, 169, 271, 274, 281, 441, 713, 901, 1156, 1871, 3025, 4894, 7921, 12817, 20736, 21319, 33551, 54289, 87842, 142129, 229969, 372100, 602071, 974169, 1576238, 2550409, 4126649, 6677056, 10803703, 17480761, 28284466, 45765225, 74049689 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The sequence contains the squares of the Fibonacci numbers (A007598(n) for n >=5).
Proof:
Let F(m) be the m-th Fibonacci number. If n = F(m)^2, n^2 - 1 = F(m)^4-1.
For m > 1, F(m)^4 - 1 = F(m-2)*F(m-1)*F(m+1)*F(m+2) with the property F(m-2) + F(m-1) + F(m+1) + F(m+2) = F(m) + F(m+3) = 2*F(m+2). (See A244855.)
F(m)^2 - 1 = F(m-1)*F(m+1) if m odd, and F(m)^2 - 1 = F(m-2)*F(m+2)if m even;
F(m)^2 + 1 = F(m-2)*F(m+2) if m odd, and F(m)^2 + 1 = F(m-1)*F(m+1) if m even, hence the product (F(m)^2 - 1)*(F(m)^2 + 1) = F(m-2)*F(m-1)*F(m+1)*F(m+2).
The primes of the sequence are 41, 103, 131, 271, 281, 1871, 21319, ...
The composites (nonsquares) of the sequence are 274, 713, 901, 4894, 12817, 33551, 87842, ...
LINKS
EXAMPLE
25^2 - 1 = 2*3*8*13 = F(5 - 2)*F(5 - 1)*F(5 + 1)*F(5 + 2) where F(5) = 5;
41^2 - 1 = 2*5*8*21;
64^2 - 1 = 3*5*13*21 = F(6 - 2)*F(6 - 1)*F(6 + 1)*F(6 + 2) where F(6) = 8;
103^2 - 1 = 3*8*13*34;
131^2 - 1 = 3*8*13*55;
169^2 - 1 = 5*8*21*34 = F(7 - 2)*F(7 - 1)*F(7 + 1)*F(7 + 2) where F(7) = 13;
271^2 - 1 = 3*5*34*144;
274^2 - 1 = 5*13*21*55;
281^2 - 1 = 2*5*8*987;
441^2 - 1 = 8*13*34*55 = F(8 - 2)*F(8 - 1)*F(8 + 1)*F(8 + 2) where F(8) = 21.
MAPLE
with(combinat, fibonacci):with(numtheory):nn:=100:lst:={}:T:=array(1..nn):
for n from 1 to nn do:
T[n]:=fibonacci(n):
od:
for p from 1 to nn-1 do:
for q from p+1 to nn-1 do:
for r from q+1 to nn-1 do:
for s from r+1 to nn-1 do:
f:=T[p]*T[q]*T[r]*T[s]+1:x:=sqrt(f):
if x=floor(x)and T[p]<>1
then
lst:=lst union {x}:
else
fi:
od:
od:
od:
od:
print(lst):
CROSSREFS
Sequence in context: A066844 A255608 A309623 * A366428 A195564 A147287
KEYWORD
nonn
AUTHOR
Michel Lagneau, Aug 14 2014
STATUS
approved

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Last modified July 22 10:19 EDT 2024. Contains 374490 sequences. (Running on oeis4.)