A209873 -id:A209873 - cp's OEIS frontend

A165559 Product of the arithmetic derivatives from 2 to n.

1, 1, 4, 4, 20, 20, 240, 1440, 10080, 10080, 161280, 161280, 1451520, 11612160, 371589120, 371589120, 7803371520, 7803371520, 187280916480, 1872809164800, 24346519142400, 24346519142400, 1071246842265600, 10712468422656000

Offset: 2

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

Cf. A003415, A190144, A209873.

A003415 := proc(n) local pfs ; if n <= 1 then 0 ; else pfs := ifactors(n)[2] ; n*add(op(2,p)/op(1,p),p=pfs) ; fi; end:
A165559 := proc(n) mul( A003415(k),k=2..n) ; end: seq( A165559(n),n=2..30) ; # R. J. Mathar, Sep 26 2009

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; a[n_] := Product[d[k], {k, 2, n}]; Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Feb 21 2014 *)

a(n) = Product_{k=2..n} A003415(k).
From Amiram Eldar, Nov 15 2020: (Start)
Sum_{n>=2} 1/a(n) = A190144.
Sum_{n>=2} (-1)^n/a(n) = A209873. (End)

Entries checked by R. J. Mathar, Sep 26 2009

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

Original entry on oeis.org

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

Paolo P. Lava, May 11 2012

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

			0.42147816129886730906200911...

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

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);

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 *)

cp's OEIS Frontend

A165559 Product of the arithmetic derivatives from 2 to n.

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