A190144 Decimal expansion of Sum_{k>=2} (1/Product_{j=2..k} j'), where n' is the arithmetic derivative of n.

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

Offset

2,1

Comments

This constant differs by 0.11320912... from the formally similar expansion of e, Sum_{n>=0} 1/n!.

Links

Table of n, a(n) for n=2..121.

Example

1/2' + 1/(2' * 3') + 1/(2' * 3' * 4') + 1/(2' * 3' * 4' * 5') + 1/(2' * 3' * 4' * 5' * 6') + ... = 1 + 1 + 1/4 + 1/4 + 1/20 + ... = 2.605072705297...

Maple

with(numtheory);
P:=proc(i)
local a, b, f, n, p, pfs;
a:=0; b:=1;
for n from 2 by 1 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2, p)/op(1, p), p=pfs);
  b:=b*f; a:=a+1/b;
od;
print(evalf(a, 300));
end:
P(1000);

Mathematica

digits = 120; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1&, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Product[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *)

Crossrefs

Cf. A003415, A190145, A190146, A190147.

Sequence in context: A274402 A115252 A108431 * A019967 A336082 A327280

Adjacent sequences: A190141 A190142 A190143 * A190145 A190146 A190147

Keyword

nonn,cons

Author

Paolo P. Lava, May 05 2011

Status

approved