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Erdős–Borwein constant

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
 2 n  −  1
,
 n   ≥   1
(sometimes called Mersenne numbers, although that name usually refers to numbers of the form
 2 p  −  1
, where
 p
is prime, but
 2 p  −  1
is either prime or composite, see A001348).
E :=
 ∞ ∑ n   = 1

 1 2 n − 1
≈ 1.606695152415291763...
A065442 Decimal expansion of Erdős–Borwein constant
 ∞

 k  = 1
 1 2 k  −  1
.
{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
 2 n  −  1, n   ≥   0
. (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

E ′ :=
 p prime ∑ p prime

 1 2 p − 1
≈ ?...,
where
 p
is prime, but
 2 p  −  1
is either prime or composite. (Obviously,
 E ′ < E
.)

A?????? Decimal expansion of sum of the reciprocals of the Mersenne numbers.

{?, ...}
A001348 Mersenne numbers:
 2 p  −  1
, where
 p
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

${\displaystyle {\begin{array}{l}\displaystyle {E{\;\;\!\!\!}{}^{\prime \prime }:=\sum _{\stackrel {\scriptstyle p{\text{ prime}}}{2^{p}-1{\text{ prime}}}}{\frac {1}{2^{p}-1}}\approx 0.516454178940788565\ldots ,}\end{array}}}$
where the primality of
 2  p  −  1
implies that
 p
is prime. (Obviously,
 E ″ < E ′ < E
.)

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
 2  p  −  1
where
 p
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

${\displaystyle {\begin{array}{l}\displaystyle {E{\;\;\!\!\!}^{\prime }-E{\;\;\!\!\!}{}^{\prime \prime }=\sum _{\stackrel {\scriptstyle p{\text{ prime}}}{2^{p}-1{\text{ composite}}}}^{}{\frac {1}{2^{p}-1}}\approx \ ?\ldots ,}\end{array}}}$
where
 p
is prime, but
 2  p  −  1
is composite.

A?????? Decimal expansion of sum of the reciprocals of the Mersenne composites.

{?, ...}
A065341 Mersenne composites:
 2  pn  −  1
is not a prime.
{2047, 8388607, 536870911, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 147573952589676412927, ...}