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 A076000 a(n) = Product_{k=1..n} k/floor(n/k). 1
 1, 1, 2, 3, 12, 20, 120, 315, 1680, 6048, 60480, 138600, 1663200, 9266400, 69189120, 340540200, 5448643200, 22870848000, 411675264000, 2111894104320, 24135932620800, 230388447744000, 5068545850368000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Sketch of proof that a(n) is an integer from Paul R. Pudaite, 9/28/2002: 1. n! = Product{p^([n/p]+[n/p^2]+...): prime p <= n}. 2. Product{[n/k]: k = 1...n} = Product{i^([n/i]-[n/i+1]): i=2...n}. 3. = Product{Product{Product{p^([n/i]-[n/i+1]): i such that p^k|i}: k such that p^k <= n}: prime p <= n}. 4. Reorganizing the exponents in the innermost product: ([n/p^k] - [n/(p^k+1)]) + ([n/(2 p^k)] - [n/(2 p^k + 1)] + ... = [n/p^k] - ([n/(p^k+1)] - [n/(2 p^k)]) - ... <= [n/p^k]. LINKS Table of n, a(n) for n=1..23. FORMULA a(n) = n!/A010786(n). EXAMPLE a(6) = 6*5*4*3*2*1/([6/1]*[6/2]*[6/3]*[6/4]*[6/5]*[6/6]) = 6!/(6*3*2*1*1*1) = 20, where [x] denotes the greatest integer <= x. MATHEMATICA Table[Product[k/Floor[n/k], {k, n}], {n, 30}] (* Harvey P. Dale, Feb 27 2013 *) PROG (PARI) a(n) = prod(k=1, n, k/(n\k)); \\ Michel Marcus, Jun 24 2021 CROSSREFS Cf. A010786, A345684. Sequence in context: A096361 A105045 A205825 * A096632 A124261 A018883 Adjacent sequences: A075997 A075998 A075999 * A076001 A076002 A076003 KEYWORD nonn AUTHOR Clark Kimberling, Sep 29 2002 STATUS approved

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Last modified August 12 19:26 EDT 2024. Contains 375113 sequences. (Running on oeis4.)