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Required fields*

Rev

Required fields*

Approximate Brun's Constant

Brun's constant is the value to which the sum of the reciprocals of twin prime pairs (1/p and 1/(p+2) where p and p+2 are both prime) converges. It is approximately 1.902160583104.

Given a positive integer N, approximate Brun's constant by summing the reciprocals of the twin prime pairs where both primes in the pair are less than N, and output the approximation.

Rules

N will be a positive integer within the representable range for your language.
The output must be as accurate as possible to the true value, within the limits of your language's floating-point implementation, ignoring any potential issues due to floating-point arithmetic inaccuracies. If your language is capable of arbitrary-precision arithmetic, it must be at least as precise as IEEE 754 double-precision arithmetic.
Alternatively, an exact fraction may be output in any consistent, unambiguous format.
If a prime appears in multiple twin prime pairs (e.g. 5, part of both (3, 5) and (5, 7)), its reciprocal contributes to the sum each time.

Test Cases

2 -> 0
6 -> 0.5333333333333333
10 -> 0.8761904761904762
13 -> 0.8761904761904762
100 -> 1.3309903657190867
620 -> 1.4999706034568274
100000 -> 1.67279958482774

Answer*

# [Jelly], <s>16</s> 14 bytes (with a little help from @Dennis)

 ’ÆRṡ2_/2+$$ÐḟFİS

[Try it online!]

While trying to improve my previous answer, I thought up a totally different algorithm, and it comes in somewhat shorter. I'm using a different post for it, as is the standard here for an answer that uses a different technique.

Dennis suggesting replacing `_/2+$$Ðḟ` with `Iċ¥Ðf2`; I'd completely forgotten about the possibility of a dyadic filter. As such, this algorithm now ties with the one that Dennis' answer used.

## Explanation

 ’ÆRṡ2Iċ¥Ðf2FİS
 ’ Decrement.
 ÆR Primes from 2 to the argument inclusive
 (i.e. 2 to the original input exclusive).
 ṡ2 Take overlapping slices of size 2.
 Ðf Keep only elements where the following is true:
 ¥ {the second parse of, which parses like this}
 Iċ 2 the differences (I) contain (ċ) 2
 F Flatten.
 İ Take 1/x {for every list element}.
 S Sum.

[Jelly]: https://github.com/DennisMitchell/jelly
[Try it online!]: https://tio.run/nexus/jelly#@/@oYebhtqCHOxcaeR7pPrT08IQ0I7cjG4L/RxvpmOkYGugYGgNJAx0zI4PYw@2Pmtb8BwA "Jelly – TIO Nexus"

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…Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one. Explanations of your answer make it more interesting to read and are very much encouraged.
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\$\begingroup\$ 2_/2+$$Ðḟ can become Iċ¥Ðf2. \$\endgroup\$

Dennis
– Dennis

2017年01月22日 02:29:31 +00:00
Commented Jan 22, 2017 at 2:29

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