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|[[Binary numeral system|Binary]]
| 1.01110100110000010101001111101100…
|-
| [[Decimal]]
| 1.45607494858268967139959535111654…
|-
| [[Hexadecimal]]
| 1.74C153ECB002353B12A0E476D3ADD…
|-
| [[Continued fraction]]
| <math>1 + \cfrac{1}{2 + \cfrac{1}{5 + \cfrac{1}{5 + \cfrac{1}{4 + \ddots}}}}</math><br><small>Note that this continued fraction is not periodic.</small>
|}
 
'''Backhouse's constant''' is a [[mathematical constant]] founded by N. Backhouse and is approximately 1.456 074 948.
 
It is defined by using the [[power series]] such that the [[coefficient]]s of successive terms are the [[prime numbers]]:
 
: <math>P(x)=1+\sum_{k=1}^\infty p_k x^k=1+2x+3x^2+5x^3+7x^4+\cdots</math>
 
and where
 
: <math>Q(x)=\frac{1}{P(x)}=\sum_{k=0}^\infty q_k x^k.</math>
 
Then:
 
: <math>\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert = 1.45607\ldots</math> {{OEIS|id=A072508}}.
 
The limit was conjectured to exist by Backhouse  which was later proved by P. Flajolet.
 
==References==
*{{mathworld|title=Backhouse's Constant|urlname=BackhousesConstant}}
*{{OEIS2C|id=A030018}}, {{OEIS2C|id=A074269}}, {{OEIS2C|id=A088751}}
 
[[Category:Mathematical constants]]
[[Category:Prime numbers]]
 
{{numtheory-stub}}

Revision as of 06:24, 23 February 2014

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