Primes in millennia

[mart_r]

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:

# 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,___

I also know that the first millennium without any prime number is 13,893,290,219,204,___

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?

[m_f_h]

You should submit this as "records in A038823" with kw "fini".

[mart_r]

Originally Posted by m_f_h

You should submit this as "records in A038823" with kw "fini".

I was thinking about it quite some time ago.
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 n-th 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.

[m_f_h]

Originally Posted by mart_r

I was thinking about it quite some time ago.
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 n-th millennium.

That's a good point. Common speak would be in favour of the latter (and btw not 0...999 but 1...1000, but since the 000 is unlikely to yield a prime, there's no problem) - and mathematically speaking there's no difference in saying [m,m+1] or [m-1,m]. Anyway, the essential thing is (IMHO) to "immortalize" the results obtained by the computational effort you invested.

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.

Originally Posted by mart_r

* 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.

Since by submitting one which is a bit less unimportant than those, you would decrease the "mean unimportance": please go ahead ! :-)

PS: I think the last argument is flawed... :-( ! But don't mind...
Let's say, a nontrivial "records" sequence for an existing sequence with A-number < 50000, which supposedly takes several hours of CPU time (7613200000 - 7613400000 took me about 10 mins) is always worth submitting.

[CRGreathouse]

Originally Posted by m_f_h

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.

Ditto. I prefer the 0th, 1st, ... 'millennia' counting primes from 1000n to 1000(n+1).

[mart_r]

Originally Posted by m_f_h

You should submit this as "records in A038823" with kw "fini".

Done.

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 )

[CRGreathouse]

Originally Posted by mart_r

I hope all terms I submitted are appearing later. Or do they generally make a cut after three lines?

They only display three lines. Other terms can be submitted as a b-file.

[mart_r]

I'm not very confident about this b-file-thingy.
With the sequence itself appearing in the database within 48 hours, I'm now waiting for more than five days for the b-file (which I think is especially important here) to show up.

Should I wait another few days or try to re-send it?

[CRGreathouse]

Give it two weeks.

[mart_r]

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 p-n(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?

[R.D. Silverman]

Originally Posted by mart_r

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 p-n(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?

My opinion is that such a thing would be a pointless waste of CPU time.