I should have also pointed out that sequence A065106 is the sequence of "Smallest Fibonacci index to produce a factor p^2 (for primes p)", so all fib(n) where n is a multiple of an element of this sequence is squareful. ----- Message from mbgreen@cis.upenn.edu --------- Date: Sun, 15 Jul 2012 17:40:11 -0400 From: mbgreen@cis.upenn.edu Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>, Richard Guy <rkg@cpsc.ucalgary.ca> Cc: "greenwald@cis.upenn.edu" <greenwald@cis.upenn.edu>
2584 is divisible by 4 (In any case, I thought that if a | fib( n ) => a | fib( kn ) for all k, so because fib( 6 ) has a square factor, all fib( 6k ) are squareful (and fib( p ) is a good place to look for big prime factors)) ----- Message from rkg@cpsc.ucalgary.ca --------- Date: Sun, 15 Jul 2012 15:11:30 -0600 (MDT) From: Richard Guy <rkg@cpsc.ucalgary.ca> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>
Dear all, None of 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, is squareful.
On Sun, 15 Jul 2012, Wouter Meeussen wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ?preferred distance? for lowish n. Wouter
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