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A190143
Decimal expansion of continued fraction 1'+1/(2'+1/(3'+1/(4'+1/(5'...)))), where n' is the arithmetic derivative of n.
1
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
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
KEYWORD
nonn,cons
AUTHOR
Paolo P. Lava, May 05 2011
STATUS
approved