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Bounds for the counting function of the Jordan-Pólya numbers. (English) Zbl 1524.11051

A Jordan-Pólya number is a product of factorials. Such integers appear naturally as orders of various permutation groups. Given such a number, its representation as a product of factorials might not be unique. For example, if \(n\) is a Jordan-Pólya number then \(n!=n\cdot(n-1)!\) already produces two representations of this Jordan-Pólya number as a product of factorials. There are a few additional sporadic solutions such as \(9!=2!3!^2 7!\) and it is conjectured that these are the only Jordan-Pólya numbers which have nonnique representations as product of factorials. In the paper under review the authors put \({\mathcal J}(x)\) for the number of Jordan-Pólya numbers up to \(x\) and prove that the estimates \[\exp((2-\varepsilon)L(x))<{\mathcal J}(x)<\exp((4+\varepsilon)L(x)\log\log L(x))\] hold for all \(\varepsilon>0\) and \(x>x_{\varepsilon}\), where \(L(x):={\sqrt{\log x}}/\log\log x\). The proofs use estimates for prime numbers and smooth numbers. For the lower bound the authors only count Jordan-Pólya numbers which are representable as products of factorials of primes as such a representation uniquely describes the concerned Jordan-Pólya number.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A41 Primes
11A51 Factorization; primality
11N05 Distribution of primes
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Online Encyclopedia of Integer Sequences:

Jordan-Polya numbers: products of factorial numbers A000142.

References:

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