Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)

[Hugo Pfoertner]

>My apologies.
>
>The problem is to find exponent E such that interval (n^E, (n+1)^E] includes a 
>prime for every n >= 1. The algorithm finds.
>
>I wrote a lazy binary search which found E = log(541)/log(52), but this value 
>appears to be incorrect.
>
>I wrote a more methodical algorithm to find E, which finds
>
>    E = log(1151)/log(95) = 1.5477771087+
>
>The algorithm shows that no exponent e < E is admissible. Also, if we compare 
>the size of the interval (n^E, (n+1)^E] to size of prime gaps listed in 
>A005250 and A002386, we find it is much larger, indicating that we will find 
>primes on (n^E, (n+1)^E] up to n^E = 2*10^16. Also, the prime gaps seem to be 
>growing more slowly than (n+1)^E-n^E, so we can have a very warm fuzzy that E 
>is the correct value.
>
>Maybe someone could verify my findings.

The decisive spot occurs at the prime 1151, which determines the least possible exponent. A description of a potential new sequence would be

floor(n^(log(1151)/log(95))) together with a remark like "there is at least one prime p in each interval fulfilling a(n) < p <= a(n+1)"

The sequence would be
    1    2    5    8   12   16   20   24   29   35   40   46   52   59   66
   73   80   87   95  103  111  119  128  136  145  154  164  173  183  193
  203  213  224  234  245  256  267  278  290  301  313  325  337  349  362
  374  387  400  413  426  439  452  466  480  493  507  522  536  550  565
  579  594  609  624  639  655  670  685  701  717  733  749  765  781  798
  814  831  848  865  882  899  916  933  951  968  986 1004 1022 1040 1058
 1076 1095 1113 1132 1151 1169 1188 1207 1226 1246 1265 1284 1304 1324 1343
 1363 1383 1403 1423 1444 1464 1485 1505 1526 1547 1567 1588 1609 1631 1652
 1673 1695 1716 1738 1760 1781 1803 1825 1848 1870 1892 1915 1937 1960 1982
 2005 2028 2051 2074 2097 2120 2144 2167 2191 2214 2238 2262 2286 2309 2334

and an example for the first occurring prime p within the intervals
  n  a(n)   p   a(n+1)
  1    1    2    2
  2    2    3    5
  3    5    7    8
  4    8   11   12
  5   12   13   16
  6   16   17   20
  7   20   23   24
  8   24   29   29
  9   29   31   35
 10   35   37   40
 11   40   41   46
 12   46   47   52
 13   52   53   59
 14   59   61   66
 15   66   67   73
 16   73   79   80
 17   80   83   87
 18   87   89   95
 19   95   97  103
 20  103  107  111
...
the decisive part:

 93 1113 1117 1132
 94 1132 1151 1151
 95 1151 1153 1169

Hugo