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

Wed Dec 12 14:27:17 CET 2007

```>>I believe it has been proved that there is always a prime between n^3
>>and (n+1)^3.  So the cubes are a considerably more compact example.
>>
>
>Legendre's conjecture says that already the squares will do the job.
>http://www.research.att.com/~njas/sequences/A000290
>http://www.research.att.com/~njas/sequences/A007491
>
>The Mathworld article http://mathworld.wolfram.com/LegendresConjecture.html
>suggests that a(n)=floor(n^(42/23)) might be a near optimal example:

Sorry for this near nonsense. Experimentally even something like a(n)=floor(n^(38/23)) will create a sequence with enough spacing for at least one prime in every interval. In this case the boundaries will be needed in some of the early intervals (e.g. 31 .. 37 )

1    3    6    9   14   19   24   31   37   44   52   60   69   78   87
97  107  118  129  141  152  165  177  190  204  217  231  246  260  275
291  306  322  339  355  372  389  407  425  443  461  480  499  519  538
558  578  599  620  641  662  684  705  728  750  773  796  819  842  866
890  914  939  964  989 1014 1039 1065 1091 1117 1144 1171 1198 1225 1252
1280 1308 1336 1365 1393 1422 1451 1481 1510 1540 1570 1601 1631 1662 1693
1724 1755 1787 1819 1851 1883 1916 1949 1982 2015 2048 2082 2116 2150 2184
2219 2253 2288 2323 2359 2394 2430 2466 2502 2538 2575 2612 2649 2686 2723
2761 2799 2837 2875 2913 2952 2991 3030 3069 3108 3148 3188 3228 3268 3308
3349 3390 3431 3472 3513 3555 3597 3639 3681 3723 3766 3808 3851 3894 3938

What would be the minimum required exponent (not theoretically, but from numerical evidence?).

Hugo

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