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prime difference function
The prime difference function is an arithmetic function for any positive integer $n$, denoted as $d_{{n}}$ and gives the difference between two consecutive primes $p_{{n}}$ and $p_{{n+1}}$:
$d_{{n}}\equiv p_{{n+1}}p_{{n}}\;.$ 
For example:

$d_{{1}}=p_{{2}}p_{{1}}=32=1$,

$d_{{10}}=p_{{11}}p_{{10}}=3129=2$,

$d_{{100}}=p_{{101}}p_{{100}}=547541=6$,

$d_{{1000}}=p_{{1001}}p_{{1000}}=79277919=8$,

$d_{{10000}}=p_{{10001}}p_{{10000}}=104743104729=14$ and so forth.
The first few values of $d_{{n}}$ for $n=1,2,3,\ldots$ are $1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,\ldots$ (OEIS A001223).
Keywords:
number theory, arithmetic function
Type of Math Object:
Definition
Major Section:
Reference
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new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
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new question: A problem about Euler's totient function by mbhatia