The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Table of n, a(n) for n=1..37. 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 Cf. A000045, A007598, A244855. Sequence in context: A066844 A255608 A309623 * A366428 A195564 A147287 Adjacent sequences: A242071 A242072 A242073 * A242075 A242076 A242077 KEYWORD nonn AUTHOR Michel Lagneau, Aug 14 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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