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A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

%I A352290 #12 Mar 17 2022 16:59:22
%S A352290 1,2,3,5,7,8,18,34,55,57,89,123,144,233,239,322,377,411,500,568,610,
%T A352290 746,788,843,987,1487,1542,1568,1636,2207,2584,2707,3173,3639,3793,
%U A352290 3804,3817,4050,4181,4217,4594,4662,5270,5778,6107,6613,8595,8972,10341,10569
%N A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.
%C A352290 A281618 is a subsequence.
%C A352290 The corresponding greatest prime Fibonacci factors of the sequence are 2, 5, 5, 13, 5, 13, 13, 89, 89, 13, 233, 89, 233, ...
%C A352290 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).
%C A352290 The Lucas numbers of the sequence are 1, 2, 3, 7, 18, 123, 322, 843, 2207, 5778, 39603, 103682, ... (subsequence of A000032).
%C A352290 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).
%e A352290 18 is in the sequence because 18^2 + 1 = 5^2*13 and 13 is a Fibonacci number.
%p A352290 q:= n-> (t-> ormap(issqr, [t+4, t-4]))(5*max(numtheory[factorset](n^2+1))^2):
%p A352290 select(q, [$1..12000])[];  # _Alois P. Heinz_, Mar 11 2022
%t A352290 With[{f = Fibonacci[Range[21]], m = f[[-1]]}, Select[Range[m], MemberQ[f, FactorInteger[#^2 + 1][[-1, 1]]] &]] (* _Amiram Eldar_, Mar 11 2022 *)
%o A352290 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
%o A352290 isok(m) = isfib(vecmax(factor(m^2+1)[,1])); \\ _Michel Marcus_, Mar 11 2022
%Y A352290 Cf. A000032, A000045, A002522, A005478, A014442, A248516, A338762, A339461.
%K A352290 nonn
%O A352290 1,2
%A A352290 _Michel Lagneau_, Mar 11 2022