Welcome to EverybodyWiki 😃 ! Log in or ➕👤 create an account to improve, watchlist or create an article like a 🏭 company page or a 👨👩 bio (yours ?)...

The Alladi–Grinstead constant is a mathematical constant that arises in the study of the factorial function, and in particular the decomposition of factorials into prime numbers.[1][2][3][4][5] It is named for the mathematicians Krishnaswami Alladi and Charles Grinstead. The decimal representation of the Alladi–Grinstead constant begins,

0.80939402054063913071793188059409131721595399242500030424202871504... (sequence A085291 in the OEIS).

The exact value of the constant can be written as the exponential of a certain infinite series expressed using the Riemann zeta function. The sequence to which the Alladi–Grinstead constant is the limit arises in the investigation of how to write n factorial as the product of n large factors, a topic studied by Paul Erdős and others.[4]

## Definition

The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers. For example, the factorial of 3 is

${\displaystyle 3!=3\cdot 2\cdot 1}$
${\displaystyle \;\;\;=3^{1}\cdot 2^{1}}$

As a less trivial example:[6]

${\displaystyle 10!=10\cdot 9!}$
${\displaystyle \;\;\;=2\cdot 5\cdot 9!}$
${\displaystyle \;\;\;=2\cdot 5\cdot 3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}}$
${\displaystyle \;\;\;=3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 2\cdot 5\cdot 5\cdot 7\cdot 2^{3}}$
${\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{3}\cdot 5\cdot 5\cdot 7\cdot 2^{3}}$
${\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 5\cdot 7\cdot 2^{3}\cdot 2^{3}}$
${\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 4\cdot 5\cdot 5\cdot 7\cdot 8\cdot 8}$

Consider all decompositions of ${\textstyle n!}$ that have length ${\textstyle n}$, and let the index ${\displaystyle i}$ label those decompositions. Then for each ${\displaystyle i}$,[7][8]

${\displaystyle n!=\prod _{j=1}^{n}p_{i,j}^{b_{i,j}},}$

where the ${\textstyle p_{i,j}}$ are prime numbers and the ${\displaystyle b_{i,j}}$ are integers greater than or equal to 1. Let

${\displaystyle \alpha (n)={\frac {\ln \max _{i}\left(p_{i,1}^{b_{i,1}}\right)}{\ln n}}.}$

As ${\textstyle n}$ tends to infinity, ${\displaystyle \alpha (n)}$ approaches a limiting value, the Alladi–Grinstead constant.[9] This constant can be written as an exponential:

${\displaystyle \lim _{n\to \infty }\alpha (n)=e^{c-1}\approx 0.80939402,}$
where ${\textstyle c}$ is given by
${\displaystyle c=\sum _{k=2}^{\infty }{\frac {1}{k}}\ln {\frac {k}{k-1}}\approx 0.78853057.}$
This constant can alternatively be expressed in terms of the Riemann zeta function,[5] as follows:
${\displaystyle c=\sum _{n=1}^{\infty }{\frac {\zeta (n+1)-1}{n}}.}$
This series for the constant ${\textstyle c}$ converges more rapidly than the one before.[5]

• Mathematical constants
• Golomb–Dickman constant