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Revision History for A304333

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A304333

Number of positive integers k such that n - L(k) is a positive squarefree number, where L(k) denotes the k-th Lucas number A000204(k).
(history; published version)

#32 by Bruno Berselli at Tue May 15 08:39:27 EDT 2018

STATUS

reviewed

approved

#31 by Peter Luschny at Tue May 15 07:11:09 EDT 2018

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proposed

reviewed

#30 by Peter Luschny at Tue May 15 07:11:04 EDT 2018

STATUS

editing

proposed

#29 by Peter Luschny at Tue May 15 07:10:42 EDT 2018

MAPLE

a := proc(n) local count, lucas, newcas;

count := 0; lucas := 1; newcas := 2;

while lucas < n do

if numtheory:-issqrfree(n - lucas) then count := count + 1 fi;

lucas, newcas := lucas + newcas, lucas;

od;

count end:

seq(a(n), n=1..90); # Peter Luschny, May 15 2018

STATUS

proposed

editing

#28 by Antti Karttunen at Sun May 13 10:45:49 EDT 2018

STATUS

editing

proposed

Discussion

Sun May 13

10:46

Antti Karttunen: [That is, what are numbers n that the whole set n - L(k) for all applicable k > 0 are squarefree numbers?]

Mon May 14

09:58

Antti Karttunen: Answering myself to the question at 10:45 (after a good night's sleep): Of course not, because differences between arbitrary Lucas numbers soon fill all values 0, 1, 2, 3 in the mod 4 set (see e.g. A066981 Number of residues of Lucas numbers modulo n), and every at least every number that is multiple of 4 is not squarefree. So no more foolish questions from me regarding this.

#27 by Antti Karttunen at Sun May 13 10:44:09 EDT 2018

COMMENTS

For all n, a(n) <= A130241(n), and for n > 14, a(n) < A130241(n) as A000204(6) = 18 is the first Lucas number that is not squarefree. - Antti Karttunen, May 13 2018

Discussion

Sun May 13

10:45

Antti Karttunen: Could we have a(n) = A130241(n) for some large n (larger than 14, I mean) ?

#26 by Antti Karttunen at Sun May 13 10:43:07 EDT 2018

COMMENTS

For all n, a(n) <= A130241(n), and for n > 14, a(n) < A130241(n) as A000204(6) = 18 is the first Lucas number that is not squarefree. - Antti Karttunen, May 13 2018

FORMULA

For all n, a(n) <= A130241(n). - Antti Karttunen, May 13 2018

STATUS

proposed

editing

#25 by Antti Karttunen at Sun May 13 10:39:49 EDT 2018

STATUS

editing

proposed

#24 by Antti Karttunen at Sun May 13 10:37:40 EDT 2018

LINKS

Antti Karttunen, <a href="/A304333/b304333.txt">Table of n, a(n) for n = 1..65537</a>

STATUS

proposed

editing

#23 by Zhi-Wei Sun at Sun May 13 10:36:45 EDT 2018

STATUS

editing

proposed