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Trying to find a pattern suggesting there are infinitely many Fibonacci primes, I consider the sequence of general term $ a_n=\omega(\prod_{k=1}^{n}F_{k}) $ where $ F_k $ is the $ k $ -th Fibonacci number and $ \omega(m) $ the number of prime factors of the largest squarefree number dividing $ m $. It seems from quick computations made by hand that $ a_{n}\sim C.n $ for some positive constant $ C $ possibly equal to $ \sqrt{2-\sqrt{2}} $ .

Can this be confirmed ? If not, can an upper bound for $ a_{n} $ be figured out ?

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Looking at Fibonacci numbers factors (gray factors are old ones, underlined factors are new ones):
$F_{1} = 1$;
$F_{2} = 1$;
$F_{3} = 2: \underline{2}$;
$F_{4} = 3: \underline{3}$;
$F_{5} = 5: \underline{5}$;
$F_{6} = 8: \color{gray}{2}^3\;\;$;
$F_{7} = 13: \underline{13}$;
$F_{8} = 21: \color{gray}{3}\;\;\underline{7}$;
$F_{9} = 34: \color{gray}{2}\;\;\underline{17}$;
$F_{10} = 55: \color{gray}{5}\;\;\underline{11}$;
$F_{11} = 89: \underline{89}$;
$F_{12} = 144: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;$;
$F_{13} = 233: \underline{233}$;
$F_{14} = 377: \color{gray}{13}\;\;\underline{29}$;
$F_{15} = 610: \color{gray}{2}\;\;\color{gray}{5}\;\;\underline{61}$;
$F_{16} = 987: \color{gray}{3}\;\;\color{gray}{7}\;\;\underline{47}$;
$F_{17} = 1597: \underline{1597}$;
$F_{18} = 2584: \color{gray}{2}^3\;\;\color{gray}{17}\;\;\underline{19}$;
$F_{19} = 4181: \underline{37}\;\;\underline{113}$;
$F_{20} = 6765: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\underline{41}$;
$F_{21} = 10946: \color{gray}{2}\;\;\color{gray}{13}\;\;\underline{421}$;
$F_{22} = 17711: \color{gray}{89}\;\;\underline{199}$;
$F_{23} = 28657: \underline{28657}$;
$F_{24} = 46368: \color{gray}{2}^5\;\;\color{gray}{3}^2\;\;\color{gray}{7}\;\;\underline{23}$;
$F_{25} = 75025: \color{gray}{5}^2\;\;\underline{3001}$;
$F_{26} = 121393: \color{gray}{233}\;\;\underline{521}$;
$F_{27} = 196418: \color{gray}{2}\;\;\color{gray}{17}\;\;\underline{53}\;\;\underline{109}$;
$F_{28} = 317811: \color{gray}{3}\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{281}$;
$F_{29} = 514229: \underline{514229}$;
$F_{30} = 832040: \color{gray}{2}^3\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\underline{31}\;\;\color{gray}{61}$;
$F_{31} = 1346269: \underline{557}\;\;\underline{2417}$;
$F_{32} = 2178309: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{47}\;\;\underline{2207}$;
$F_{33} = 3524578: \color{gray}{2}\;\;\color{gray}{89}\;\;\underline{19801}$;
$F_{34} = 5702887: \color{gray}{1597}\;\;\underline{3571}$;
$F_{35} = 9227465: \color{gray}{5}\;\;\color{gray}{13}\;\;\underline{141961}$;
$F_{36} = 14930352: \color{gray}{2}^4\;\;\color{gray}{3}^3\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\underline{107}$;
$F_{37} = 24157817: \underline{73}\;\;\underline{149}\;\;\underline{2221}$;
$F_{38} = 39088169: \color{gray}{37}\;\;\color{gray}{113}\;\;\underline{9349}$;
$F_{39} = 63245986: \color{gray}{2}\;\;\color{gray}{233}\;\;\underline{135721}$;
$F_{40} = 102334155: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{7}\;\;\color{gray}{11}\;\;\color{gray}{41}\;\;\underline{2161}$;
$F_{41} = 165580141: \underline{2789}\;\;\underline{59369}$;
$F_{42} = 267914296: \color{gray}{2}^3\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{211}\;\;\color{gray}{421}$;
$F_{43} = 433494437: \underline{433494437}$;
$F_{44} = 701408733: \color{gray}{3}\;\;\underline{43}\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{307}$;
$F_{45} = 1134903170: \color{gray}{2}\;\;\color{gray}{5}\;\;\color{gray}{17}\;\;\color{gray}{61}\;\;\underline{109441}$;
$F_{46} = 1836311903: \underline{139}\;\;\underline{461}\;\;\color{gray}{28657}$;
$F_{47} = 2971215073: \underline{2971215073}$;
$F_{48} = 4807526976: \color{gray}{2}^6\;\;\color{gray}{3}^2\;\;\color{gray}{7}\;\;\color{gray}{23}\;\;\color{gray}{47}\;\;\underline{1103}$;
$F_{49} = 7778742049: \color{gray}{13}\;\;\underline{97}\;\;\underline{6168709}$;
$F_{50} = 12586269025: \color{gray}{5}^2\;\;\color{gray}{11}\;\;\underline{101}\;\;\underline{151}\;\;\color{gray}{3001}$;
$F_{51} = 20365011074: \color{gray}{2}\;\;\color{gray}{1597}\;\;\underline{6376021}$;
$F_{52} = 32951280099: \color{gray}{3}\;\;\color{gray}{233}\;\;\color{gray}{521}\;\;\underline{90481}$;
$F_{53} = 53316291173: \underline{953}\;\;\underline{55945741}$;
$F_{54} = 86267571272: \color{gray}{2}^3\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{53}\;\;\color{gray}{109}\;\;\underline{5779}$;
$F_{55} = 139583862445: \color{gray}{5}\;\;\color{gray}{89}\;\;\underline{661}\;\;\underline{474541}$;
$F_{56} = 225851433717: \color{gray}{3}\;\;\color{gray}{7}^2\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\color{gray}{281}\;\;\underline{14503}$;
$F_{57} = 365435296162: \color{gray}{2}\;\;\color{gray}{37}\;\;\color{gray}{113}\;\;\underline{797}\;\;\underline{54833}$;
$F_{58} = 591286729879: \underline{59}\;\;\underline{19489}\;\;\color{gray}{514229}$;
$F_{59} = 956722026041: \underline{353}\;\;\underline{2710260697}$;
$F_{60} = 1548008755920: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{31}\;\;\color{gray}{41}\;\;\color{gray}{61}\;\;\underline{2521}$;
$F_{61} = 2504730781961: \underline{4513}\;\;\underline{555003497}$;
$F_{62} = 4052739537881: \color{gray}{557}\;\;\color{gray}{2417}\;\;\underline{3010349}$;
$F_{63} = 6557470319842: \color{gray}{2}\;\;\color{gray}{13}\;\;\color{gray}{17}\;\;\color{gray}{421}\;\;\underline{35239681}$;
$F_{64} = 10610209857723: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{47}\;\;\underline{1087}\;\;\color{gray}{2207}\;\;\underline{4481}$;
$F_{65} = 17167680177565: \color{gray}{5}\;\;\color{gray}{233}\;\;\underline{14736206161}$;
$F_{66} = 27777890035288: \color{gray}{2}^3\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{9901}\;\;\color{gray}{19801}$;
$F_{67} = 44945570212853: \underline{269}\;\;\underline{116849}\;\;\underline{1429913}$;
$F_{68} = 72723460248141: \color{gray}{3}\;\;\underline{67}\;\;\color{gray}{1597}\;\;\color{gray}{3571}\;\;\underline{63443}$;
$F_{69} = 117669030460994: \color{gray}{2}\;\;\underline{137}\;\;\underline{829}\;\;\underline{18077}\;\;\color{gray}{28657}$;
$F_{70} = 190392490709135: \color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{71}\;\;\underline{911}\;\;\color{gray}{141961}$;
$F_{71} = 308061521170129: \underline{6673}\;\;\underline{46165371073}$;
$F_{72} = 498454011879264: \color{gray}{2}^5\;\;\color{gray}{3}^3\;\;\color{gray}{7}\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{23}\;\;\color{gray}{107}\;\;\underline{103681}$;
$F_{73} = 806515533049393: \underline{9375829}\;\;\underline{86020717}$;
$F_{74} = 1304969544928657: \color{gray}{73}\;\;\color{gray}{149}\;\;\color{gray}{2221}\;\;\underline{54018521}$;
$F_{75} = 2111485077978050: \color{gray}{2}\;\;\color{gray}{5}^2\;\;\color{gray}{61}\;\;\color{gray}{3001}\;\;\underline{230686501}$;
$F_{76} = 3416454622906707: \color{gray}{3}\;\;\color{gray}{37}\;\;\color{gray}{113}\;\;\color{gray}{9349}\;\;\underline{29134601}$;
$F_{77} = 5527939700884757: \color{gray}{13}\;\;\color{gray}{89}\;\;\underline{988681}\;\;\underline{4832521}$;
$F_{78} = 8944394323791464: \color{gray}{2}^3\;\;\underline{79}\;\;\color{gray}{233}\;\;\color{gray}{521}\;\;\underline{859}\;\;\color{gray}{135721}$;
$F_{79} = 14472334024676221: \underline{157}\;\;\underline{92180471494753}$;
$F_{80} = 23416728348467685: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{7}\;\;\color{gray}{11}\;\;\color{gray}{41}\;\;\color{gray}{47}\;\;\underline{1601}\;\;\color{gray}{2161}\;\;\underline{3041}$;
$F_{81} = 37889062373143906: \color{gray}{2}\;\;\color{gray}{17}\;\;\color{gray}{53}\;\;\color{gray}{109}\;\;\underline{2269}\;\;\underline{4373}\;\;\underline{19441}$;
$F_{82} = 61305790721611591: \color{gray}{2789}\;\;\color{gray}{59369}\;\;\underline{370248451}$;
$F_{83} = 99194853094755497: \underline{99194853094755497}$;
$F_{84} = 160500643816367088: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{83}\;\;\color{gray}{211}\;\;\color{gray}{281}\;\;\color{gray}{421}\;\;\underline{1427}$;
$F_{85} = 259695496911122585: \color{gray}{5}\;\;\color{gray}{1597}\;\;\underline{9521}\;\;\underline{3415914041}$;
$F_{86} = 420196140727489673: \underline{6709}\;\;\underline{144481}\;\;\color{gray}{433494437}$;
$F_{87} = 679891637638612258: \color{gray}{2}\;\;\underline{173}\;\;\color{gray}{514229}\;\;\underline{3821263937}$;
$F_{88} = 1100087778366101931: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{43}\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{263}\;\;\color{gray}{307}\;\;\underline{881}\;\;\underline{967}$;
$F_{89} = 1779979416004714189: \underline{1069}\;\;\underline{1665088321800481}$;
$F_{90} = 2880067194370816120: \color{gray}{2}^3\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{31}\;\;\color{gray}{61}\;\;\underline{181}\;\;\underline{541}\;\;\color{gray}{109441}$;
$F_{91} = 4660046610375530309: \color{gray}{13}^2\;\;\color{gray}{233}\;\;\underline{741469}\;\;\underline{159607993}$;
$F_{92} = 7540113804746346429: \color{gray}{3}\;\;\color{gray}{139}\;\;\color{gray}{461}\;\;\underline{4969}\;\;\color{gray}{28657}\;\;\underline{275449}$;

we can notice that almost each Fibonacci number "brings" one (or even more) new factor(s) to main prime factor list.

As one can see, the ratio $\dfrac{a_n}{n}$ grows slowly with $n$, and for $n=37$ the product $\prod_{k=1}^nF_n$ will have $38$ factors: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 61, 73, 89, 107, 109, 113, 149, 199, 233, 281, 421, 521, 557, 1597, 2207, 2221, 2417, 3001, 3571, 19801, 28657, 141961, 514229;$
So, for $n=37$ we have $\dfrac{a_n}{n}>1$.

And from this list we can derive that for $n=92$ we have $\dfrac{a_n}{n}\approx 1.40217$.

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