A075999
Product{[n/k + 1/2]: k=1,2,...,n}, where [x + 1/2] denotes the integer nearest to x.
Original entry on oeis.org
1, 2, 6, 8, 30, 72, 112, 384, 1080, 1800, 6336, 13824, 26208, 94080, 345600, 368640, 1057536, 5598720, 6566400, 24192000, 74511360, 93671424, 457850880, 955514880, 2156544000, 5887365120, 16460236800, 40967700480, 101025792000
Offset: 1
Cf.
A075998 (product of second-nearest integers),
A010786 (product of [n/k], k=1, 2, ..., n).
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Table[Product[Floor[n/k+1/2],{k,n}],{n,30}] (* Harvey P. Dale, Feb 15 2015 *)
A076000
a(n) = Product_{k=1..n} k/floor(n/k).
Original entry on oeis.org
1, 1, 2, 3, 12, 20, 120, 315, 1680, 6048, 60480, 138600, 1663200, 9266400, 69189120, 340540200, 5448643200, 22870848000, 411675264000, 2111894104320, 24135932620800, 230388447744000, 5068545850368000
Offset: 1
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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Table[Product[k/Floor[n/k],{k,n}],{n,30}] (* Harvey P. Dale, Feb 27 2013 *)
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a(n) = prod(k=1, n, k/(n\k)); \\ Michel Marcus, Jun 24 2021
Original entry on oeis.org
1, 3, 4, 30, 24, 168, 288, 1080, 1920, 12672, 8640, 131040, 161280, 414720, 774144, 10575360, 8294400, 36771840, 69672960, 383201280, 729907200, 4578508800, 2043740160, 45287424000, 65213890560, 180592312320, 347807416320
Offset: 1
a(6) = (1/2)*<6/1>*<6/2>*<6/3>*<6/4>*<6/5>*<6/6> = (1/2)*7*4*3*1*2*2 = 168, where <x> denotes the integer second-nearest to x, as defined at A075998.
Cf.
A075998 (product of second-nearest integers),
A010786 (product of [n/k], k=1, 2, ..., n).
A331213
a(n) = 1 + Sum_{i=1..n} (-1)^i * Product_{j=1..i} floor(n/j).
Original entry on oeis.org
1, 0, 1, -2, 5, -4, 13, -27, 89, -80, 191, -450, 2365, -1182, 3221, -13034, 40433, -22320, 96373, -193761, 772981, -728930, 1599357, -3428425, 21411337, -13595724, 31407273, -110011850, 377746853, -198079308, 1096983421, -2241234465, 7565512161, -6472208192
Offset: 0
a(4) = 1 - 4 + 4*floor(4/2) - 4*floor(4/2)*floor(4/3) + 4*floor(4/2)*floor(4/3)*floor(4/4) = 1 - 4 + 4*2 - 4*2*1 + 4*2*1*1 = 5.
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[1] cat [1+&+[(-1)^i*(&*[Floor(n/j):j in [1..i]]):i in [1..n]]:n in [1..33]]; // Marius A. Burtea, Jan 13 2020
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a[n_] := 1 + Sum[(-1)^i * Product[Floor[n/j], {j, 1, i}],{i, 1, n}]; Array[a, 34, 0] (* Amiram Eldar, Jan 13 2020 *)
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{a(n) = 1+sum(i=1, n, (-1)^i*prod(j=1, i, floor(n/j)))}
A145119
a(n) = Product_{k=1..n-1} (ceiling(n/k) - ceiling(n/k) mod 2).
Original entry on oeis.org
1, 2, 4, 16, 32, 96, 384, 1024, 2048, 10240, 30720, 73728, 294912, 688128, 1835008, 12582912, 25165824, 56623104, 283115520, 629145600, 1887436800, 11072962560, 26575110144, 57982058496, 231928233984, 753766760448, 1758789107712
Offset: 1
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