A075742 Fibonacci numbers for which both the value and index are the product of an odd number of distinct primes; that is, numbers Fibonacci(k) for which mu(k) = mu(Fibonacci(k)) = -1.
2, 5, 13, 89, 233, 1597, 28657, 514229, 24157817, 433494437, 2971215073, 44945570212853, 190392490709135, 99194853094755497, 1500520536206896083277, 3928413764606871165730, 1066340417491710595814572169, 19134702400093278081449423917
Offset: 1
Examples
11 is a prime and Fibonacci(11) = 89 is a prime, 13 is a prime and Fibonacci(13) = 233 is a prime, but Fibonacci(16) = 987 = 3*7*47 and 16 is not squarefree and 30 = 2*3*5 is the product of an odd number of distinct primes but Fibonacci(30) = 832040 = 2^3*5*11*31*61 is not squarefree, ...
Programs
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Maple
with(combinat, fibonacci): m2_supM_fib := proc(n); if (numtheory[mobius](n)=-1) then if (numtheory[mobius](fibonacci(n))=-1) then RETURN(fibonacci(n)); fi; fi; end:
Extensions
Incorrect 83621143489848422977 removed and more terms from Sean A. Irvine, Mar 05 2025