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A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number. 1
1, 2, 3, 5, 7, 8, 18, 34, 55, 57, 89, 123, 144, 233, 239, 322, 377, 411, 500, 568, 610, 746, 788, 843, 987, 1487, 1542, 1568, 1636, 2207, 2584, 2707, 3173, 3639, 3793, 3804, 3817, 4050, 4181, 4217, 4594, 4662, 5270, 5778, 6107, 6613, 8595, 8972, 10341, 10569 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A281618 is a subsequence.
The corresponding greatest prime Fibonacci factors of the sequence are 2, 5, 5, 13, 5, 13, 13, 89, 89, 13, 233, 89, 233, ...
The Fibonacci numbers of the sequence are 1, 2, 3, 5, 8, 34, 55, 89, 144, 233, 377, 610, 987, 2584, 4181, 10946, 17711, ... (subsequence of A000045).
The Lucas numbers of the sequence are 1, 2, 3, 7, 18, 123, 322, 843, 2207, 5778, 39603, 103682, ... (subsequence of A000032).
The prime numbers of the sequence are 2, 3, 5, 7, 89, 233, 239, 1487, 2207, 2707, 3793, 4217, 11789, 11981, 13763, ... including the prime Fibonacci numbers 2, 3, 5, 89, 233, 1066340417491710595814572169, ... (subsequence of A005478).
LINKS
EXAMPLE
18 is in the sequence because 18^2 + 1 = 5^2*13 and 13 is a Fibonacci number.
MAPLE
q:= n-> (t-> ormap(issqr, [t+4, t-4]))(5*max(numtheory[factorset](n^2+1))^2):
select(q, [$1..12000])[]; # Alois P. Heinz, Mar 11 2022
MATHEMATICA
With[{f = Fibonacci[Range[21]], m = f[[-1]]}, Select[Range[m], MemberQ[f, FactorInteger[#^2 + 1][[-1, 1]]] &]] (* Amiram Eldar, Mar 11 2022 *)
PROG
(PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = isfib(vecmax(factor(m^2+1)[, 1])); \\ Michel Marcus, Mar 11 2022
CROSSREFS
Sequence in context: A328724 A039892 A278645 * A350707 A285282 A105404
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 11 2022
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)