Prime gaps of increasing merit.

[Ed Pegg Jr]

Prime gaps of increasing merit = (p_{n+1}-p_n)/log(p_n) might be worthwhile.
The last term was recently found by Siegfried Herzog & Tomás Oliveira e 
Silva,
and is the first increase found in six years.

2, 1, 1.4427
3, 2, 1.82048
7, 4, 2.05559
113, 14, 2.96147
1129, 22, 3.12985
1327, 34, 4.72835
19609, 52, 5.26116
31397, 72, 6.95352
155921, 86, 7.19238
360653, 96, 7.50254
370261, 112, 8.73501
1357201, 132, 9.34782
4652353, 154, 10.0307
2010733, 148, 10.197
17051707, 180, 10.8097
20831323, 210, 12.4615
191912783, 248, 13.003
436273009, 282, 14.1753
2300942549, 320, 14.8447
3842610773, 336, 15.2247
4302407359, 354, 15.9586
10726904659, 382, 16.5396
25056082087, 456, 19.0441
304599508537, 514, 19.4386
461690510011, 532, 19.8078
1346294310749, 582, 20.839
1408695493609, 588, 21.0198
1968188556461, 602, 21.266
2614941710599, 652, 22.8034
13829048559701, 716, 23.6633
19581334192423, 766, 25.0281
218209405436543, 906, 27.4408
1693182318746371, 1132, 32.2825
804212830686677669, 1442, 34.9757.

http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm#top20merit

--Ed Pegg Jr