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Maximal factorial dividing product

Source: 2024 Israel TST Test 8 P1

May 10, 2024
factorialnumber theoryfloor function

Problem Statement

For each positive integer nn let ana_n be the largest positive integer satisfying (an)!∣∏k=1n⌊nk⌋(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right. Show that there are infinitely many positive integers mm for which am+1<ama_{m+1}<a_m.
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