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: 1

Author

Paolo P. Lava, May 05 2011

Keywords

• nonn
• cons

Comments

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

Examples

			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...
		

Crossrefs

Cf. A003415, A190145, A190146, A190147.

Programs

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 *)