[seqfan] Re: On some constellations of primes

[zak seidov]

Smallest set of 13 (VladSh's) consecutive primes:
s=prime(1785277..1785289)={28751809, 28751851, 28751857, 28751873, 28751893, 28751903, 28751929, 28751941, 28751969, 28751977, 28752007, 28752019, 28752037},
12 corresponding smallest primes q(k) between (1/2)s(k) and (1/2)s(k+1):
q(k=1..12)={14375923, 14375927, 14375929, 14375939, 14375947, 14375957, 
 14375969, 14375981, 14375987, 14376001, 14376007, 14376013};
 indices of these primes q:
 {932933, 932934, 932935, 932936, 932937, 932938, 932939, 932940, 932942, 932943, 932944, 932945} 
 that is in all cases (except for 8th interval ) there is only one intermedate prime q;
 in the 8th interval (28751941/2, 28751969/2) there are 2 primes q: 14375981 and 14375983.

 I suggest that longer sets are possible.

Disclaimer: no warranty though I guess it's OK,
Zak Seidov

PS I have several sets of lengths 11, 12 and 13, in case of any interest.


----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> To: seqfan at list.seqfan.eu
> Cc: 
> Sent: Friday, October 5, 2012 9:06 PM
> Subject: [seqfan] On some constellations of primes
> 
> Dear SeqFans,
> 
> The following sequence of 11 consecutive primes
> 55469,55487,55501,55511,55529,55541,55547,55579,55589,55603,55609
> possesses an interesting property: between every adjacent half-primes there 
> exists at least one prime. In particular, between the first two half-primes 
> there are 3 primes: 27737,27739,27743. This sequence is connected with two my 
> sequences:  A166251 and A182426 (see a theorem in comment). I conjectured that 
> there exist arbitrary long sequences of consecutive primes with such property. 
> Reinhard Zumkeller extended A182426 up to n=10000 and after that (private 
> communication) up to n=50000, but I did not find a term>11. Thus till now I 
> do not know a longer sequence than the above mentioned one. Can anyone find a 
> longer sequence or indicate
> a new bound N, such that up to N such a sequence not exists.
> 
> Regards,
> Vladimir
> 
> Shevelev Vladimir‎
> 
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