A190143 Decimal expansion of continued fraction 1'+1/(2'+1/(3'+1/(4'+1/(5'...)))), where n' is the arithmetic derivative of n.

5, 4, 6, 6, 8, 0, 3, 2, 4, 4, 6, 6, 4, 2, 3, 3, 7, 1, 4, 6, 2, 1, 7, 2, 7, 2, 4, 7, 0, 9, 5, 0, 0, 2, 7, 2, 2, 6, 3, 0, 2, 5, 6, 8, 9, 8, 5, 2, 7, 9, 2, 1, 6, 6, 1, 4, 5, 4, 3, 4, 9, 8, 1, 2, 6, 4, 9, 0, 4, 2, 0, 4, 8, 4, 0, 8, 5, 9, 4, 9, 0, 0, 9, 7, 3, 2, 6, 7, 0, 6, 5, 1, 6, 0, 3, 7, 3, 0, 3, 0, 4, 0, 6, 8, 2, 1, 6, 0, 7, 7, 4, 4, 8, 4, 2, 9, 0, 5, 8

Offset

0,1

Comments

Good approximation up to the 9th decimal digit with 22948/41977.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Example

0.546680324466423...

Maple

with(numtheory);
P:=proc(i)
local a, f, n, p, pfs;
a:=0;
for n from i by -1 to 1 do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2, p)/op(1, p), p=pfs);
  a:=1/(a+f);
od;
print(evalf(1/a, 1000));
end:
P(10000);

Mathematica

digits = 119; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; f[m_] := f[m] = Fold[d[#2] + 1/#1 &, 1, Range[m] // Reverse] // RealDigits[#, 10, digits] & // First; f[digits]; f[m = 2 digits]; While[f[m] != f[m/2], m = 2 m]; f[m] (* Jean-François Alcover, Feb 21 2014 *)

Crossrefs

Cf. A003415.

Sequence in context: A328263 A138255 A153451 * A089687 A251792 A159894

Adjacent sequences: A190140 A190141 A190142 * A190144 A190145 A190146

Keyword

nonn,cons

Author

Paolo P. Lava, May 05 2011

Status

approved