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Alladi-Grinstead Constant

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Consider decomposition of the factorial n! into multiplicative factors p_k^(b_k) arranged in nondecreasing order. For example,

4!=3·2^3
(1)
=2·3·4
(2)
=2·2·2·3
(3)

and

5!=3·5·2^3
(4)
=2·3·2^2·5
(5)
=2·2·2·3·5.
(6)

The numbers of such partitions for n=2, 3, ... are 1, 1, 3, 3, 10, 10, 30, 75, 220, ... (OEIS A085288).

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

9!=2·2·2·2·2·2^2·5·7·3^4
(7)
=2·2·2·2·3·5·7·2^3·3^3
(8)
=2·2·2·2·5·7·2^3·3^2·3^2
(9)
=2·2·2·3·2^2·2^2·5·7·3^3
(10)
=2·2·2·2^2·2^2·5·7·3^2·3^2
(11)
=2·2·2·3·3·5·7·3^2·2^4
(12)
=2·2·3·3·2^2·5·7·2^3·3^2
(13)
=2·2·3·3·3·3·5·7·2^5
(14)
=2·3·3·2^2·2^2·2^2·5·7·3^2
(15)
=2·3·3·3·3·2^2·5·7·2^4
(16)
=2·3·3·3·3·5·7·2^3·2^3
(17)
=3·3·3·3·2^2·2^2·5·7·2^3.
(18)

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

Now let

m(n)=max(p_1^(b_1)),
(19)

i.e., m(n) is the least prime factor raised to its appropriate power in the factorization of length n. For n=4, 5, ..., m(n) 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

alpha(n)=(lnm(n))/(lnn)
(20)

where ln(x) is the natural logarithm. Therefore, for the case n=9, m(9)=3 and

alpha(9)=(ln3)/(ln9)=(ln3)/(2ln3)=1/2.
(21)
Alladi-GrinsteadConstant

For large n, alpha(n) approaches a constant

lim_(n->infty)alpha(n)=e^(c-1)
(22)
=0.80939402054...
(23)

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

c=sum_(k=2)^(infty)1/kln(k/(k-1))
(24)
=0.7885305659115...
(25)

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

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

c=sum_(k=2)^(infty)k^(-2)+1/2k^(-3)+1/3k^(-4)+...
(26)
=sum_(k=2)^(infty)sum_(n=1)^(infty)1/(nk^(n+1)).
(27)

Interchanging the order of summation then gives

c=sum_(n=1)^(infty)sum_(k=2)^(infty)1/(nk^(n+1))
(28)
=sum_(n=1)^(infty)(zeta(n+1)-1)/n,
(29)

where zeta(n) is the Riemann zeta function.

c can also be expressed as the integral

c=int_0^1ln|_1/x_|dx.
(30)

See also

Factorial

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References

Alladi, K. and Grinstead, C. "On the Decomposition of n! 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 n as the Product of n 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."

Referenced on Wolfram|Alpha

Alladi-Grinstead Constant

Cite this as:

Weisstein, Eric W. "Alladi-Grinstead Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Alladi-GrinsteadConstant.html

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