Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)
Hugo Pfoertner
all at abouthugo.de
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.
>>
>>Franklin T. Adams-Watters
>
>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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