Prime Numbers and the relationship between n and P_n

Feb 2013
38
1
London, England, UK
We often hear it said that either there is no relation between the natural, or counting numbers, $n$, and their counterparts the primes, $P_{n}$, or that if there is, it is so recondite as to be completely obscure. This is something of a sacred cow, but I don't believe it to be true. I'm not claiming the following is an exact relationship by any means, but it is a start, and a step in the right direction; it holds pretty well for the first 100,000 primes with remarkable accuracy:

2*$n$*$P_{n}$ $\approx$ $P_{n^2}$

Examples:

$n$----------$P_{n}$-------2*$n$*$P_{n}$-------$P_{n^2}$

1-----------2------------4------------2

2-----------3------------12-----------7

3-----------5------------30-----------23

4-----------7------------56-----------53

5-----------11-----------110----------97

6-----------13-----------156----------151

7-----------17-----------238----------227

8-----------19-----------304----------311

9-----------23-----------414----------419

10----------29-----------580----------541

11----------31-----------682----------661

12----------37-----------888----------827

13----------41-----------1066---------1009

14----------43-----------1204---------1193

15----------47-----------1410---------1427

16----------53-----------1696---------1619

17----------59-----------2006---------1879

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77----------389----------59906--------58579

78----------397----------61932--------60289

79----------401----------63358--------62081

80----------409----------65440--------63809

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