A190147 Decimal expansion of Sum{k=1..infinity}(1/Sum{j=1..k} j^j’), where n’ is the arithmetic derivative of n.

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

Offset

1,2

Links

Table of n, a(n) for n=1..120.

Example

1/1^1’+1/(1^1’+2^2’)+1/(1^1’+2^2’+3^3’)+1/(1^1’+2^2’+3^3’+4^4’)+... = 1+1/3+1/6+1/262+... = 1.50781066762289...

Maple

with(numtheory);
P:=proc(i)
local a, b, f, n, p, pfs;
a:=0; b:=0;
for n from 1 by 1 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2, p)/op(1, p), p=pfs);
  b:=b+n^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/Sum[j^d[j], {j, 1, k}], {k, 1, 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, A190144, A190145, A190146.

Sequence in context: A180660 A221711 A200400 * A346190 A108745 A114124

Adjacent sequences: A190144 A190145 A190146 * A190148 A190149 A190150

Keyword

nonn,cons

Author

Paolo P. Lava, May 05 2011

Status

approved