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Alladi–Grinstead constant

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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[edit]

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

As a less trivial example:[6]


Consider all decompositions of that have length , and let the index label those decompositions. Then for each ,[7][8]

where the are prime numbers and the are integers greater than or equal to 1. Let

As tends to infinity, approaches a limiting value, the Alladi–Grinstead constant.[9] This constant can be written as an exponential:

where is given by
This constant can alternatively be expressed in terms of the Riemann zeta function,[5] as follows:
This series for the constant converges more rapidly than the one before.[5]

See also[edit]

  • Mathematical constants
  • Golomb–Dickman constant

References[edit]

  1. Weisstein, Eric. "Alladi-Grinstead Constant". mathworld.wolfram.com. Retrieved 2017-05-01.
  2. Alladi, Krishnaswami; Grinstead, Charles. "On the decomposition of n! into prime powers". Journal of Number Theory. 9 (4): 452–458. doi:10.1016/0022-314x(77)90006-3.
  3. Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 120–122. ISBN 978-0521818056. Search this book on Amazon.com Logo.png
  4. 4.0 4.1 Guy, Richard K. (1994). "Factorial n as the Product of n Large Factors". Unsolved Problems in Number Theory. Springer-Verlag. p. 79. ISBN 978-0387208602. Search this book on Amazon.com Logo.png
  5. 5.0 5.1 5.2 Weisstein, Eric. "Convergence Improvement". mathworld.wolfram.com. Retrieved 2017-05-03.
  6. (sequence A085290 in the OEIS)
  7. (sequence A085288 in the OEIS)
  8. (sequence A085289 in the OEIS)
  9. (sequence A085291 in the OEIS)

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