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The
Erdős–Borwein constant, named after
Paul Erdős and Peter Borwein,
^{[1]} is the sum of the
reciprocals of the
almost powers of two, i.e. numbers of the form
,
(sometimes called Mersenne numbers, although that name usually refers to numbers of the form
, where
is prime, but
is either prime or composite, see
A001348).

E := ≈ 1.606695152415291763... 
A065442 Decimal expansion of Erdős–Borwein constant
.

{1, 6, 0, 6, 6, 9, 5, 1, 5, 2, 4, 1, 5, 2, 9, 1, 7, 6, 3, 7, 8, 3, 3, 0, 1, 5, 2, 3, 1, 9, 0, 9, 2, 4, 5, 8, 0, 4, 8, 0, 5, 7, 9, 6, 7, 1, 5, 0, 5, 7, 5, 6, 4, 3, 5, 7, 7, 8, ...}
A000225 . (Sometimes called Mersenne numbers, although that name is usually reserved for
A001348.)

{0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, ...}
Sum of the reciprocals of the Mersenne numbers
The sum of the reciprocals of the Mersenne numbers is

where
is
prime, but
is either prime or composite. (Obviously,
.)
A?????? Decimal expansion of sum of the reciprocals of the Mersenne numbers.

{?, ...}
A001348 Mersenne numbers:
, where
is prime.

{3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911, 2147483647, 137438953471, 2199023255551, 8796093022207, 140737488355327, ...}
Sum of the reciprocals of the Mersenne primes
The sum of the reciprocals of the Mersenne primes is
 ${\begin{array}{l}\displaystyle {E{\;\;\!\!\!}{}^{\prime \prime }:=\sum _{\stackrel {\scriptstyle p{\text{ prime}}}{2^{p}1{\text{ prime}}}}{\frac {1}{2^{p}1}}\approx 0.516454178940788565\ldots ,}\end{array}}$
where the primality of
implies that
is prime. (Obviously,
.)
A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

{5, 1, 6, 4, 5, 4, 1, 7, 8, 9, 4, 0, 7, 8, 8, 5, 6, 5, 3, 3, 0, 4, 8, 7, 3, 4, 2, 9, 7, 1, 5, 2, 2, 8, 5, 8, 8, 1, 5, 9, 6, 8, 5, 5, 3, 4, 1, 5, 4, 1, 9, 7, 0, 1, 4, 4, 1, 9, ...}
A000668 Mersenne primes (of form
where
is a prime).

{3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, ...}
Sum of the reciprocals of the Mersenne composites
The sum of the reciprocals of the Mersenne composites is
 ${\begin{array}{l}\displaystyle {E{\;\;\!\!\!}^{\prime }E{\;\;\!\!\!}{}^{\prime \prime }=\sum _{\stackrel {\scriptstyle p{\text{ prime}}}{2^{p}1{\text{ composite}}}}^{}{\frac {1}{2^{p}1}}\approx \ ?\ldots ,}\end{array}}$
where
is
prime, but
is composite.
A?????? Decimal expansion of sum of the reciprocals of the Mersenne composites.

{?, ...}
A065341 Mersenne composites:
is not a prime.

{2047, 8388607, 536870911, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 147573952589676412927, ...}
Notes