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Swiss primes: This plot shows what I am calling Swiss prime numbers up to 30 million, shown in magenta open circles. I am sure there is a mathematical definition for these prime numbers but I haven't found it yet. They are the prime numbers that are equidistant from their two neighboring primes. In this case, the "loneliness," defined as sqrt(distance_lower * distance_upper) (the geometric mean of separations) is an integer and is equal to the distance from the lower or upper neighboring prime. As you can see from the magenta plot, with a logarithmic x-axis, the loneliness of the Swiss primes also grows faster than log(prime number), as shown by the positive curvature of the envelope of the magneta circles. If you know the proper name for these "Swiss primes" please let me know! See also my previous post: https://plus.google.com/110146523961072500429/posts/TsSWSprcYaH

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- That above link to 29998259 on number empire (amazing by the way) says that it is the 1,857,756th prime. I found that it is the 57,243th balanced prime. So, the fraction of primes <30million that are balanced is 0.0308 or about 3%.May 3, 2012
- the fraction of numbers <30million that are prime is 0.0619 or about 6% (1857859/30000000). And the fraction of numbers <=30million that are balanced primes is 0.0019 or 0.1%. That seems like a lot.May 3, 2012
- The maximum isolation of a balanced prime less than 30,000,000 is 72 (=2*2*2*3*3), which occurs 4 times: 23,346,809; 25,443,361; 28,933,939; and 29,747,071. For 23,346,809, 72 is 4.24 times its natural log.May 3, 2012
- Besides the balanced prime 5 (between 3 and 7), the isolation or loneliness of a balanced prime appears to have to be a multiple of 6 (see graph above with magenta at 2, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72). That must mean something.May 3, 2012
- Hmm, I think this explains the origin of "6": http://en.wikipedia.org/wiki/Primes_in_arithmetic_progression So three consecutive primes in arithmetic progression would be CPAP-3 and has a minimum separation of 3# = 3 * 2 = 6. But that doesn't show why the separation is a multiple of 6, does it?May 3, 2012
- From Andre Nicolas on http://math.stackexchange.com/questions/147101/is-there-a-theorem-or-conjecture-that-specifies-a-balanced-prime-must-be-multipl

He points out that it's easy to probe that primes in arithmetic progression (AP-k) must be a MULTIPLE of k#. Thus, for a balanced prime, the separation must be a multiple of 6. This is easy to reason out, as one of the math.stackexchange answerers described.May 19, 2012