login
A190144
Decimal expansion of Sum_{k>=2} (1/Product_{j=2..k} j'), where n' is the arithmetic derivative of n.
6
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!.
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
KEYWORD
nonn,cons
AUTHOR
Paolo P. Lava, May 05 2011
STATUS
approved