20150519, 12:17  #1 
"Andrew Booker"
Mar 2013
92_{10} Posts 
OEIS A071580: Smallest prime of the form k*a(n1)*a(n2)*...*a(1)+1
I have been working on extending OEIS sequence A071580, whose nth term is the smallest prime congruent to 1 mod the product of all smaller terms. The sequence enjoys several nice properties in common with the sequence of Mersenne primes:
1) There is a fast algorithm to prove primality: since p1 has a prime factor of about square root size, the simplest variant of Pocklington's criterion runs about as quickly as a Fermat PRP test. 2) It has rapid growth (doubly exponential), so we get to big primes quickly. 3) It has an intrinsic definition, as opposed to (say) Proth primes, which are a bit quicker to test, but for which the choice of numbers is completely ad hoc. I have computed the first 23 terms, the last of which is a prime with over a million digits. I used gwnum to find the terms and doublechecked the first 22 of them with gmp. It took a lot of effort to find the 23rd term (it was roughly a factor of 3 larger than expected), and I'm thinking of stopping at this point, but if any of you guys with substantial resources would be interested in helping to find the 24th term or doublecheck the 23rd, let me know. There is a reasonable chance of finding the 24th term (which will be a prime with over 2 million digits) within a year. 
20150519, 19:03  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·7·359 Posts 
That is a nice sequence. And good thinking about converting it into the kseries (A258081); one needs a compact representation for a simple sieve and whatnot.
Technically, this series is not as fast as Mersenne's (wall clock wise), because for all terms generic mod reduction on a generic FFT size will be used (making the proof at least twice slower than a similarsized Mp or a GFN or a cyclotomic GenUni number). My bet for a vizaviz Mersenne contender for the largest known prime is between the latter two classes; all they need is a GIMPS' size crowd to follow. I'll double check your terms with PFGW and scripts. I'll start with writing a quickanddirty sieve, maybe tonight. 
20150519, 19:11  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×7×359 Posts 
Also, A071580(22) and A071580(23) should be in the UTM prime database (even if they are not minimal, pending a double check,  they are still great primes)!
Email Chris Caldwell and send him a zip of the whole sequence, because it will be a helper file for proving successive terms. 
20150519, 20:54  #4  
"Andrew Booker"
Mar 2013
2^{2}×23 Posts 
Quote:
Quote:


20150519, 21:05  #5  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×7×359 Posts 
I sent Chris the zip file with the (regenerated by me) sequence.
That's all he needs. Here's what I wrote: Quote:


20150519, 21:10  #6  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·7·359 Posts 
Quote:
I've warmed up by brute force doublechecking up to a(18) and a couple more are pending, but if you send me the blueprints, that would save even more time for DC. I'll PM you. 

20150519, 22:02  #7 
"Andrew Booker"
Mar 2013
2^{2}·23 Posts 

20150521, 05:59  #8 
Jun 2003
5·1,087 Posts 
The smaller of the two has now appeared in Top5000 (http://primes.utm.edu/primes/page.php?id=119934). Interesting things to note:
It has an current rank higher than entry rank (as of this writing)! It took PFGW 4 passes to prove this (where the N+1 pass should take about 2x N1 pass)! BTW, the sister sequence (https://oeis.org/A258081) is empty? Can someone post all the k values here as a handy reference? EDIT: Reconstructed k values from the Top 5000 submission data Code:
1, 1, 1, 1, 2, 10, 12, 10, 21, 25, 70, 670, 239, 2115, 586, 1619, 26800, 2505, 99019, 40903, 285641, 67166 Last fiddled with by axn on 20150521 at 06:28 
20150521, 06:13  #9 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·7·359 Posts 
It is actually not empty  and if you log in (at top right), then you can see any draft of any sequence (inc. this). Handy!
Even if you are not entering sequences you can create a Wiki account.But in any case the sequence is this ATM: Code:
1, 1, 1, 1, 2, 10, 12, 10, 21, 25, 70, 670, 239, 2115, 586, 1619, 26800, 2505, 99019, 40903, 285641, 67166, 1852765 P.S. What may be the case is that C.C. is running all ladder proofs, first. (And that is the right way to do it. I ran them all, too.) He is rigorous, and even if he wasn't, David Broadhurst is even more rigorous. Last fiddled with by Batalov on 20150521 at 06:26 Reason: tpyos 
20150521, 11:49  #10 
Jun 2003
5·1,087 Posts 
Thanks Serge. I missed your reply as I was editing mine to include the k values. I will register at OEIS (not sure... I may already have an account there).
Also found the "" side http://oeis.org/A090475. How far has this one been extended? EDIT: Now the Top5000 entry has its entrance rank adjusted, so everything is fine. Last fiddled with by axn on 20150521 at 11:51 
20150521, 17:54  #11  
"Andrew Booker"
Mar 2013
1011100_{2} Posts 
Quote:
I do have just under 10k terms of A061092, but we'll never get to top 5000 primes with that (current size is ~41k digits). 

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