Dirichlet divisor problem and Factorial summation limit

Clear[c, cc, n, m];
Sum[Sum[1, {k, Floor[n/(m + 1)] + 1, Floor[n/m - 1/m]}], {n, 1, nn}]
cc = 6
Table[Sum[
  Sum[Sum[1, {k, Floor[n/(m + 1)] + 1, Floor[n/m - 1/m]}], {n, 1,
    c  }], {m, 1, c}], {c, 1, cc + 10}]
Table[Sum[
  Sum[Sum[1, {k, Floor[n/(m + 1)] + 1, Floor[n/m - 1/m]}], {n, 1,
    c!}], {m, 1, 2*Floor[(c - 1)/2] + 1}], {c, 1, cc}]
Table[-(1/4 c! (2 + c!) (-2 + 1/(1 + Floor[1/2 (-1 + c)])) +
    c! HarmonicNumber[1 + 2 Floor[1/2 (-1 + c)]]), {c, 1, cc}]

cc = 200
Show[ListLinePlot[
  table = (Table[Sum[DivisorSigma[0, k], {k, n}], {n, 3, cc}] -
     Table[c*((1 +
           Log[-(1/(4) c! (2 + c!) (-2 + 1/(
                    1 + Floor[1/2 (-1 + c)])) +
                 c! HarmonicNumber[1 + 2 Floor[1/2 (-1 + c)]])]/c/
            2) + 2*EulerGamma - 1), {c, 3, cc}])]]
ListLinePlot[Re[Accumulate[table] - 0/4*(Range[Length[table]])]]
ListLinePlot[Re[Accumulate[table] - 1/4*(Range[Length[table]])]]
ListLinePlot[
 Accumulate[Re[Accumulate[table] - 1/4*(Range[Length[table]])]]]



"start"
nn = 150;
Show[ListLinePlot[
  Table[Sum[If[And[n > 1*k, n < 2*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[2/1] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 2*k, n < 3*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[3/2] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 3*k, n < 4*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[4/3] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 4*k, n < 5*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[5/4] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 5*k, n < 6*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[6/5] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 6*k, n < 7*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[7/6] - 1, {n, 1, nn}],
  PlotStyle -> Red]]
Show[ListLinePlot[
  Table[Sum[If[And[n > 7*k, n < 8*k], n/k, 0], {k, 1, n}], {n, 1,
    nn}]], ListLinePlot[Table[n*Log[8/7] - 1, {n, 1, nn}],
  PlotStyle -> Red]]

ListLinePlot[
 Table[Sum[If[And[n > 1*k, n < 2*k], n/k, 0], {k, 1, n}] -
   n*Log[2/1] + 1, {n, 1, nn}]]
ListLinePlot[
 Table[Sum[If[And[n > 2*k, n < 3*k], n/k, 0], {k, 1, n}] -
   n*Log[3/2] + 1, {n, 1, nn}]]
ListLinePlot[
 Table[Sum[If[And[n > 3*k, n < 4*k], n/k, 0], {k, 1, n}] -
   n*Log[4/3] + 1, {n, 1, nn}]]
ListLinePlot[
 Table[Sum[If[And[n > 4*k, n < 5*k], n/k, 0], {k, 1, n}] -
   n*Log[5/4] + 1, {n, 1, nn}]]
ListLinePlot[
 Table[Sum[If[And[n > 5*k, n < 6*k], n/k, 0], {k, 1, n}] -
   n*Log[6/5] + 1, {n, 1, nn}]]

ListLinePlot[
 Accumulate[
   Table[Sum[If[And[n > 1*k, n < 2*k], n/k, 0], {k, 1, n}] -
     n*Log[2/1] + 1, {n, 1, nn}]] - 1/8]
ListLinePlot[
 Accumulate[
   Table[Sum[If[And[n > 2*k, n < 3*k], n/k, 0], {k, 1, n}] -
     n*Log[3/2] + 1, {n, 1, nn}]] - 7/10]
ListLinePlot[
 Accumulate[
   Table[Sum[If[And[n > 3*k, n < 4*k], n/k, 0], {k, 1, n}] -
     n*Log[4/3] + 1, {n, 1, nn}]] - 1]
ListLinePlot[
 Accumulate[
   Table[Sum[If[And[n > 4*k, n < 5*k], n/k, 0], {k, 1, n}] -
     n*Log[5/4] + 1, {n, 1, nn}]] - 3/2]
ListLinePlot[
 Accumulate[
   Table[Sum[If[And[n > 5*k, n < 6*k], n/k, 0], {k, 1, n}] -
     n*Log[6/5] + 1, {n, 1, nn}]]*2]
"end"