Erdős–Borwein constant

This article page is a stub, please help by expanding it.

The Erdős–Borwein constant, named after Paul Erdős and Peter Borwein, 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

${\begin{array}{l}\displaystyle {E{\;\;\!\!\!}{}^{\prime \prime }:=\sum _{\stackrel {\scriptstyle p{\text{ prime}}}{2^{p}-1{\text{ prime}}}}{\frac {1}{2^{p}-1}}\approx 0.516454178940788565\ldots ,}\end{array}}$

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

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

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, ...}