A210937 Decimal expansion of the continued fraction 1'+1/(2'+2/(3'+3/...)), where n' is the arithmetic derivative of n.

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

Offset

0,1

Comments

A good approximation up to the 9th decimal digit is 4796/11379.

References

1

Links

Table of n, a(n) for n=0..128.

Example

0.42147816129886730906200911...

Maple

with(numtheory);
A210937:= proc(n)
local a, b, c, I, p, pfs;
b:=1;
for i from n by -1 to 2 do
  pfs:=ifactors(i)[2]; a:=i*add(op(2, p)/op(1, p), p=pfs); b:=1/b*a+i;
od;
print(evalf(b, 500));
end:
A210937(10000);

Mathematica

digits = 129; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1&, FactorInteger[n], {1}]]; f[m_] := f[m] = Fold[d[#2]+#2/#1&, 1, Range[m] // Reverse] // RealDigits[#, 10, digits]& // First; f[digits]; f[m = 2digits]; While[f[m] != f[m/2], m = 2m]; f[m] (* Jean-François Alcover, Feb 21 2014 *)

Crossrefs

Cf. A003415, A190144, A190145, A190146, A190147, A209873.

Sequence in context: A153727 A349245 A356631 * A016506 A346995 A228132

Adjacent sequences: A210934 A210935 A210936 * A210938 A210939 A210940

Keyword

nonn,cons

Author

Paolo P. Lava, May 11 2012

Status

approved