Prime Puzzio: Mean Prime Gaps

There are exactly 9 gaps and 18 composites below 29 (the 10th prime number). Since 9 divides 18, we can say the average (or mean) gap less than 29 is 2, i.e., an integer.

Here is a list of record breaking mean gaps which are integers:

  n                   nth prime             mean gap
  ------------------  ------------------    -----------------------------
  2                   3                      0 [Honaker, May 2000]
  10                  29                     2  "
  68                  337                    4  "
  438                 3061                   6 [Kok Seng Chua, May 2000]
  2616                23537                  8  "
  100350              1304539               12  "
  637198              9557957               14  "
  27066970            514272413             18  "
  179992840           3779849621            20  "
  55762149072         1505578024919         26 [Carmody, July 2003]
  382465573492        11091501631241        28  "
  2636913002950       81744303091421        30  "
  126979448983454     4444280714420857      34 [J. K. Andersen, April 2008]
  885992692751466     32781729631804207     36  "
  43525513764814972   1784546064357413813   40  "
  306268030480171300  13169525310647365859  42  "
  ------------------  --------------------  -----------------------------

Update: Phil Carmody has provided proof that odd mean gaps are impossible.  Note that the mean gap g occurs about exp(g + 2).