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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)

 (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.

Factorial

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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."