A345683
a(n) = n! * Sum_{k=1..n} 1/floor(n/k).
Original entry on oeis.org
1, 3, 14, 66, 444, 2880, 25080, 216720, 2247840, 24071040, 304335360, 3752179200, 54965433600, 810550540800, 13176376012800, 219079045785600, 4078723532083200, 75227891042304000, 1550619342784512000, 31871016307113984000, 710529031487987712000, 16180987966182014976000
Offset: 1
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Table[n! * Sum[1/Floor[n/k], {k, 1, n}], {n, 1, 25}]
Table[n!*(Sum[(Floor[n/j] - Floor[n/(j + 1)])/j, {j, 1, n}]), {n, 1, 25}]
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a(n) = n!*sum(k=1, n, 1/(n\k)); \\ Michel Marcus, Jun 24 2021
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k))/(1-x))) \\ Seiichi Manyama, Jul 23 2022
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from math import factorial, isqrt
def A345683(n): return (m:=factorial(n))*(n-1)+m//n+sum((q:=n//k)*(m//k-m//(k-1))+m//q for k in range(2,isqrt(n)+1)) # Chai Wah Wu, Oct 27 2023
A345682
a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).
Original entry on oeis.org
1, 2, 7, 26, 148, 804, 6228, 47424, 441936, 4288320, 50437440, 560373120, 7723935360, 106618256640, 1614841401600, 25127582054400, 446784010444800, 7727747269939200, 152873884406476800, 2966599550251008000, 62987912790921216000, 1378192085174919168000
Offset: 1
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Table[n! * Sum[1/(k*Floor[n/k]), {k, 1, n}], {n, 1, 25}]
Table[n! * Sum[(HarmonicNumber[Floor[n/j]] - HarmonicNumber[Floor[n/(1 + j)]])/j, {j, 1, n}], {n, 1, 25}]
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a(n) = n!*sum(k=1, n, 1/(k*(n\k))); \\ Michel Marcus, Jun 24 2021
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k)/k)/(1-x))) \\ Seiichi Manyama, Jul 23 2022
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
A342933
a(n) = n! * Sum_{k=1..n} k^2/floor(n/k).
Original entry on oeis.org
1, 9, 80, 654, 6264, 59760, 665160, 7565040, 96929280, 1286046720, 18976083840, 286363123200, 4775047200000, 81792956044800, 1515077749785600, 28898470215014400, 594066352700620800, 12467555729620992000, 280797084422959104000, 6460327992512249856000, 157769680941941612544000
Offset: 1
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Table[n!*Sum[k^2/Floor[n/k], {k, 1, n}], {n, 1, 25}]
Table[n!*Sum[(Floor[n/j]*(1 + Floor[n/j])*(1 + 2*Floor[n/j]) - Floor[n/(1 + j)]*(1 + Floor[n/(1 + j)])*(1 + 2*Floor[n/(1 + j)]))/6/j, {j, 1, n}], {n, 1, 25}]
Showing 1-4 of 4 results.
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