Alladi-Grinstead Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $N(n)$ be the number of ways in which the Factorial $n!$ can be decomposed into $n$ Factors of the form ${p_k}^{b_k}$ arranged in nondecreasing order. Also define

$$m(n)\equiv {\rm max}({p_1}^{b_1}),$$ (1)

i.e., $m(n)$ is the Least Prime Factor raised to its appropriate Power in the factorization. Then define

$$\alpha(n)\equiv {\ln m(n)\over\ln n}$$ (2)

where $\ln(x)$ is the Natural Logarithm. For instance,

$$9! = 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2^2\cdot 5\cdot 7\cdot 3^4$$

$$= 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 5\cdot 7\cdot 2^3\cdot 3^3$$

$$= 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot 7\cdot 2^3\cdot 3^2\cdot 3^2$$

$$= 2\cdot 2\cdot 2\cdot 3\cdot 2^2\cdot 2^2\cdot 5\cdot 7\cdot 3^3$$

$$= 2\cdot 2\cdot 2\cdot 2^2\cdot 2^2\cdot 5\cdot 7\cdot 3^2\cdot 3^2$$

$$= 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5\cdot 7\cdot 3^2\cdot 2^4$$

$$= 2\cdot 2\cdot 3\cdot 3\cdot 2^2\cdot 5\cdot 7\cdot 2^3\cdot 3^2$$

$$= 2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 3\cdot 5\cdot 7\cdot 2^5$$

$$= 2\cdot 3\cdot 3\cdot 2^2\cdot 2^2\cdot 2^2\cdot 5\cdot 7\cdot 3^2$$

$$= 2\cdot 3\cdot 3\cdot 3\cdot 3\cdot 2^2\cdot 5\cdot 7\cdot 2^4$$

$$= 2\cdot 3\cdot 3\cdot 3\cdot 3\cdot 5\cdot 7\cdot 2^3\cdot 2^3$$

$$= 3\cdot 3\cdot 3\cdot 3\cdot 2^2\cdot 2^2\cdot 5\cdot 7\cdot 2^3,$$ (3)

so

$$\alpha(9)={\ln 3\over \ln 9}={\ln 3\over 2\ln 3}={1\over 2}.$$ (4)

For large $n$,

$$\lim_{n\to\infty} \alpha(n)=e^{c-1}=0.809394020534\ldots,$$ (5)

where

$$c\equiv \sum_{k=2}^\infty {1\over k}\ln\left({k\over k-1}\right).$$ (6)

References

Alladi, K. and Grinstead, C. ``On the Decomposition of $n!$ into Prime Powers.'' J. Number Th. 9, 452-458, 1977.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/aldgrns/aldgrns.html

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