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:

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${d}_{1}={p}_{2}{p}_{1}=32=1$,

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${d}_{10}={p}_{11}{p}_{10}=3129=2$,

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${d}_{100}={p}_{101}{p}_{100}=547541=6$,

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${d}_{1000}={p}_{1001}{p}_{1000}=79277919=8$,

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${d}_{10000}={p}_{10001}{p}_{10000}=104743104729=14$ and so forth.
The first few values of ${d}_{n}$ for $n=1,2,3,\mathrm{\dots}$ are $1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,\mathrm{\dots}$ (http://www.research.att.com/ njas/sequences^{}/eisA.cgi?Anum=001223OEIS A001223).
Title  prime difference function 

Canonical name  PrimeDifferenceFunction 
Date of creation  20130322 12:49:16 
Last modified on  20130322 12:49:16 
Owner  XJamRastafire (349) 
Last modified by  XJamRastafire (349) 
Numerical id  6 
Author  XJamRastafire (349) 
Entry type  Definition 
Classification  msc 11N05 
Classification  msc 11A25 
Classification  msc 11A41 