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Search: id:A076000
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| A076000 |
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Product_{k=1..n} k/[n/k]. |
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+0
1
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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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].
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EXAMPLE
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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.
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CROSSREFS
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Cf. n!/A010786(n).
Sequence in context: A067391 A096361 A105045 this_sequence A096632 A124261 A077755
Adjacent sequences: A075997 A075998 A075999 this_sequence A076001 A076002 A076003
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Sep 29 2002
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