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Alladi-Grinstead Constant

المؤلف:  Alladi, K. and Grinstead, C.

المصدر:  "On the Decomposition of n! into Prime Powers." J. Number Th. 9

الجزء والصفحة:  ...

1-10-2020

1181

Alladi-Grinstead Constant

Consider decomposition the factorial  into multiplicative factors  arranged in nondecreasing order. For example,

			(1)
			(2)
			(3)

and

			(4)
			(5)
			(6)

The numbers of such partitions for , 3, ... are 1, 1, 3, 3, 10, 10, 30, 75, 220, ... (OEIS A085288).

Now consider the number of such decompositions that are of length . For instance,

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

The numbers of such partitions for , 3, ... are 0, 0, 1, 1, 2, 2, 5, 12, 31, 31, 78, 78, 191, ... (OEIS A085289).

Now let

(19)

i.e.,  is the least prime factor raised to its appropriate power in the factorization of length . For , 5, ...,  is given by 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, ... (OEIS A085290).

Finally, define

(20)

where  is the natural logarithm. Therefore, for the case ,  and

(21)

For large ,  approaches a constant

			(22)
			(23)

(OEIS A085291), known as the Alladi-Grinstead constant, where

			(24)
			(25)

(OEIS A085361). The constant  is also associated with so-called alternating Lüroth representations (Finch 2003, p. 62).

The series for  can be transformed to one with much better convergence properties by expanding the addend about infinity to get

			(26)
			(27)

Interchanging the order of summation then gives

			(28)
			(29)

where  is the Riemann zeta function.

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REFERENCES:

Alladi, K. and Grinstead, C. "On the Decomposition of  into Prime Powers." J. Number Th. 9, 452-458, 1977.

Finch, S. R. "Alladi-Grinstead Constant." §2.9 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 120-122, 2003.

Guy, R. K. "Factorial  as the Product of  Large Factors." §B22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 79, 1994.

Sloane, N. J. A. Sequences A085288, A085289, A085290, A085291, and A085361 in "The On-Line Encyclopedia of Integer Sequences."