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Fibonorial

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The fibonorial n!_F, also called the Fibonacci factorial, is defined as

n!_F=product_(k=1)^nF_k,

where F_k is a Fibonacci number. For n=1, 2, ..., the first few fibonorials are 1, 1, 2, 6, 30, 240, 3120, 65520, ... (OEIS A003266).

The fibonorials are asymptotic to

n!_F∼C(phi^(n(n+1)/2))/(5^(n/2)),

where C is the Fibonacci factorial constant and phi is the golden ratio.

The first few values of n such that n!_F-1 is prime are given by 4, 5, 6, 7, 8, 14, 15, ... (OEIS A059709), with the next index, if it exists, exceeding 1300 (M. S. Branicky, Aug. 17, 2024).

The first few values of n such that n!_F+1 is prime are given by 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (OEIS A053408), with the next index, if it exists, exceeding 1100 (R. Price, May 27, 2019).

See also

Factorial, Fibonacci Factorial Constant, Fibonacci Number, Fibonomial Coefficient, Integer Sequence Primes, Primorial

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References

Brousseau, A. Fibonacci and Related Number Theoretic Tables. San Jose, CA: Fibonacci Association, p. 69, 1972.Finch, S. R. "Fibonacci Factorials." §1.2.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 10, 2003.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 597, 1994.Matiyasevich, Y. V. and Guy, R. K. "A New Formula for Pi." Amer. Math. Monthly 93, 631-635, 1986.Sloane, N. J. A. Sequences A003266/M1692, A053408, and A059709 in "The On-Line Encyclopedia of Integer Sequences."

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Fibonorial

Cite this as:

Weisstein, Eric W. "Fibonorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fibonorial.html

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