20090222, 12:49  #1 
Dec 2008
you know...around...
28A_{16} Posts 
Primes in millennia
I had spent some time looking for the successive record minimum number of primes in a "millennium", i.e. intervals between ...000 and ...999.
This is how far I've gotten: Code:
# of pr. millennium 168 0,___ 135 1,___ 127 2,___ 120 3,___ 119 4,___ 114 5,___ 107 7,___ 106 10,___ 103 11,___ 102 14,___ 98 16,___ 94 18,___ 92 29,___ 90 38,___ 88 40,___ 85 43,___ 80 64,___ 76 88,___ 73 168,___ 71 180,___ 69 212,___ 68 293,___ 67 356,___ 63 452,___ 61 555,___ 59 638,___ 58 871,___ 54 913,___ 53 1,637,___ 52 2,346,___ 46 3,279,___ 43 7,176,___ 42 14,420,___ 38 15,369,___ 36 36,912,___ 35 51,459,___ 34 96,733,___ 33 113,376,___ 31 141,219,___ 28 200,315,___ 27 233,047,___ 26 729,345,___ 25 951,847,___ 24 1,704,275,___ 23 1,917,281,___ 22 2,326,985,___ 21 2,937,877,___ 20 6,973,534,___ 18 7,362,853,___ 17 12,838,437,___ 16 26,480,476,___ 15 34,095,574,___ 13 162,661,473,___ 12 304,552,694,___ 10 378,326,417,___ 9 1,252,542,156,___ 8 3,475,851,270,___ 7 6,603,973,861,___ 6 7,613,200,181,___ So I thought if I never ask, I may never find out about the gap  the first millennia with 5, 4, 3, 2 primes and one prime respectively. Is someone out there who can provide this information? Last fiddled with by mart_r on 20090222 at 12:50 
20090222, 15:12  #3  
Dec 2008
you know...around...
2·5^{2}·13 Posts 
Quote:
Yet 1) there are also some other sequences I still like to submit but I think they are too unimportant * and 2) I'm a bit unsure about what numbers I should choose: 0, 1, 2, 3, 4, 5, 7, 10 ... as the constant digits of the interval or 1, 2, 3, 4, 5, 6, 8, 11 ... as it being the nth millennium. * Though there are so many "unimportant" sequences in the database compared to others that aren't in there which I think make it unnecessarily unwieldy. 

20090222, 15:24  #4  
Feb 2007
2^{4}·3^{3} Posts 
Quote:
PS1: On a second thought, "record indices of A038823" (in decreasing sense: n such that A038823(n) is lower than all A038823(k) with k<n), would correspond to the former choice ; I think this is preferrable. Quote:
PS: I think the last argument is flawed... :( ! But don't mind... Let's say, a nontrivial "records" sequence for an existing sequence with Anumber < 50000, which supposedly takes several hours of CPU time (7613200000  7613400000 took me about 10 mins) is always worth submitting. Last fiddled with by m_f_h on 20090222 at 15:39 Reason: several reasons... 

20090222, 17:48  #5 
Aug 2006
1011101100001_{2} Posts 
Ditto. I prefer the 0th, 1st, ... 'millennia' counting primes from 1000n to 1000(n+1).

20090222, 19:29  #6  
Dec 2008
you know...around...
1212_{8} Posts 
Quote:
I hope all terms I submitted are appearing later. Or do they generally make a cut after three lines? (by the way  look at this comparison: there are 13 primes between 162,661,473,001 and 162,661,474,000, but 70 primes between 166,007,963,001 and 166,007,964,000 ) 

20090222, 19:52  #7 
Aug 2006
3^{2}·5·7·19 Posts 
They only display three lines. Other terms can be submitted as a bfile.
Last fiddled with by CRGreathouse on 20090222 at 19:52 Reason: fix spacing 
20090302, 17:50  #8 
Dec 2008
you know...around...
2·5^{2}·13 Posts 
I'm not very confident about this bfilethingy.
With the sequence itself appearing in the database within 48 hours, I'm now waiting for more than five days for the bfile (which I think is especially important here) to show up. Should I wait another few days or try to resend it? 
20090302, 18:09  #9 
Aug 2006
3^{2}·5·7·19 Posts 
Give it two weeks.

20090304, 19:16  #10 
Dec 2008
you know...around...
2×5^{2}×13 Posts 
Yep, it's updated now.
Another thing I'd like to share is "primes in a quadratic pattern", i.e. p ± n*(n+1) for a run of n's (say up to 12 or 13, starting from 0). I've searched p+n(n+1) to 37*10^12 some time ago and am now with pn(n+1), currently at ~ 4*10^12. Strangely, the OEIS only lists a sequence for n=0...6, and nothing else. That would require me to send about 18 separate sequences. I was thus thinking about a single sequence for each p+ and p, listing ten (or more) terms successively for each n. Any opinions/suggestions? 
20090304, 19:30  #11  
Nov 2003
1D24_{16} Posts 
Quote:


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