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:

• $d_1=p_2-p_1=3-2=1$,

• $d_{10}=p_{11}-p_{10}=31-29=2$,

• $d_{100}=p_{101}-p_{100}=547-541=6$,

• $d_{1000}=p_{1001}-p_{1000}=7927-7919=8$,

• $d_{10000}=p_{10001}-p_{10000}=104743-104729=14$ 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).