If there are k composites preceding Nth prime,

then the proposed merit is

k / (log N)^2

There are 17 known gaps with merit > 1:

merit k N

1.15817 3 5

1.13821 33 218

1.13709 1131 49749629143527

1.10242 13 31

1.07487 71 3386

1.05228 209 1319946

1.05033 455 1094330260

1.03633 905 6822667965941

1.03522 147 149690

1.03436 111 31546

1.03257 765 662221289044

1.02282 1441 20004097201301080

1.01896 651 94906079601

1.01838 463 1820471369

1.01650 1475 34952141021660496

1.00560 1369 10570355884548335

1.00197 113 40934

More data there:

http://www.primefan.ru/stuff/primes/gaps/gaps.xls
Screenshot:

http://www.primefan.ru/stuff/primes/gaps/gaps.gif
Cram�r/Granville suggest that only finite

number of gaps has merit larger than

2/exp(eulergamma) = 1.12291896713377...

Currently only three such gaps are known:

from 7 to 11, from 1327 to 1361 and

from 1693182318746371 to 1693182318747503.

Maybe no more exist, who knows?

Best regards,

Andrey

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