A303384 Total area of all rectangles with dimensions s and t where s | t, n = s + t and s <= t.
0, 1, 2, 7, 4, 22, 6, 35, 26, 50, 10, 126, 12, 86, 100, 155, 16, 247, 18, 294, 172, 182, 22, 590, 124, 242, 260, 518, 28, 860, 30, 651, 364, 386, 380, 1365, 36, 470, 484, 1390, 40, 1532, 42, 1134, 1144, 662, 46, 2542, 342, 1395, 772, 1526, 52, 2380, 788
Offset: 1
Links
Programs
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GAP
List([1..60],n->Sum([1..Int(n/2)],i->i*(n-i)*(Int((n-i)/i)-Int((n-i-1)/i)))); # Muniru A Asiru, Jun 07 2018
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Magma
[0] cat [&+[k*(n-k)*((n-k) div k)-(n-k-1) div k: k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Jun 07 2018
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Maple
with(numtheory): seq(n*sigma(n) - sigma[2](n), n=1..60); # Ridouane Oudra, Apr 15 2021
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Mathematica
Table[Sum[i (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[n/2]}], {n, 80}] a[n_] := n * DivisorSigma[1, n] - DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Dec 11 2023 *) -
PARI
a(n) = sum(i=1, n\2, i*(n-i)*((n-i)\i - (n-i-1)\i)); \\ Michel Marcus, Jun 07 2018
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PARI
a(n) = sumdiv(n, d, d*(n-d)); \\ Daniel Suteu, Jun 19 2018
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PARI
a(n) = {my(f = factor(n)); n * sigma(f) - sigma(f, 2);} \\ Amiram Eldar, Dec 11 2023
Formula
a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * (floor((n-i)/i) - floor((n-i-1)/i)).
a(n) = Sum_{d|n} d*(n-d). - Daniel Suteu, Jun 19 2018
a(n) = n*sigma(n) - sigma_2(n). - Ridouane Oudra, Apr 15 2021
From Amiram Eldar, Dec 11 2023: (Start)
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) - zeta(3) = 0.442877... . (End)