A113767
I am interesting in extending the series A113767
[url]https://oeis.org/A113767[/url] I had done some work on this in the past [url]http://www.mersenneforum.org/showthread.php?t=18347[/url] 1) How do I submit the two terms found 6071, 218431? 2) Is there an efficient sieve for these forms? 3) Would these sequences be finite? 
[QUOTE=Citrix;485085]I am interesting in extending the series A113767
[url]https://oeis.org/A113767[/url] I had done some work on this in the past [url]http://www.mersenneforum.org/showthread.php?t=18347[/url] 1) How do I submit the two terms found 6071, 218431? 2) Is there an efficient sieve for these forms? 3) Would these sequences be finite?[/QUOTE] 1) get an OEIS account, ( or use somebody who has one) to edit the sequence. 2) modular remainders are inefficient but give that if a(n1) is 2 mod 3, k is odd, if a(n1) is 1 mod 3,k is even.( Oops math for the related sequence) 3) no clue. 
1) I have figured out how to add the terms. Thanks.
2) I am looking for efficient software (not the math). This is basically a fixed k sieve (but the k is very large). 
It feels like these sequences ought to be finite. The only longterm modular restriction I can think of for members of the sequence is that they cannot be 1 mod p for any p other than 2 and the previous term (correct me if I've missed something!). Since 78557 has covering set {3, 5, 7, 13, 19, 37, 73}, 78557+3*5*7*13*19*37*73*n is a Sierpinski number for every n, and 78557 is not 1 mod any of the primes in its covering set, so eventually the sequence ought to hit a number of this form, if it does not hit another Sierpinski number first.

[QUOTE=10metreh;485138]It feels like these sequences ought to be finite. The only longterm modular restriction I can think of for members of the sequence is that they cannot be 1 mod p for any p other than 2 and the previous term (correct me if I've missed something!). Since 78557 has covering set {3, 5, 7, 13, 19, 37, 73}, 78557+3*5*7*13*19*37*73*n is a Sierpinski number for every n, and 78557 is not 1 mod any of the primes in its covering set, so eventually the sequence ought to hit a number of this form, if it does not hit another Sierpinski number first.[/QUOTE]
It can also hit a cunningham chain of first or second kind respectively though. 
Is someone able to modify srsieve to sieve for (((((((((((((((((1*2^1+1)*2^1+1)*2^2+1)*2^1+1)*2^5+1)*2^1+1)*2^1+1)*2^29+1)*2^3+1)*2^37+1)*2^31+1)*2^227+1)*2^835+1)*2^115+1)*2^7615+1)*2^6071+1)*2^218431+1)*2^n+1?

[QUOTE=Citrix;485204]Is someone able to modify srsieve to sieve for (((((((((((((((((1*2^1+1)*2^1+1)*2^2+1)*2^1+1)*2^5+1)*2^1+1)*2^1+1)*2^29+1)*2^3+1)*2^37+1)*2^31+1)*2^227+1)*2^835+1)*2^115+1)*2^7615+1)*2^6071+1)*2^218431+1)*2^n+1?[/QUOTE]
I think your best bet might be to modify mtsieve adding a fixed k sieve with k being a gmp number. Your runtime might be dominated though by calculating k % p as that is a very large k. Whatever you do I don't think it is going to be overly fast. [url]http://www.mersenneforum.org/rogue/mtsieve.html[/url] 
[QUOTE=science_man_88;485140]It can also hit a cunningham chain of first or second kind respectively though.[/QUOTE]
But this doesn't affect the finiteness or otherwise of the sequence, as Cunningham chains are finite. Also, as the terms get larger, the probability of hitting a Cunningham chain should tend to 0 (by the prime number theorem), whereas the probability of hitting a Sierpinski number does not. 
We can make the list of all primes:
a(m,n+1) is the smallest prime of the form a(m,n)*2^k+1 (k>=0) a(m+1,1) is the smallest prime not in the first m rows [CODE] 2 3 7 29 59 1889 3779 7559 5 11 23 47 47*2^583+1 13 53 107 857 6857 17 137 1097 140417 [/CODE] The first row of this table is indeed A084435. For the Riesel analog: a(m,n+1) is the smallest prime of the form a(m,n)*2^k1 (k>=1) a(m+1,1) is the smallest prime not in the first m rows [CODE] 2 3 5 19 37 73 9343 7 13 103 823 1685503 11 43 5503 17 67 2143 [/CODE] 
[QUOTE=henryzz;485211]I think your best bet might be to modify mtsieve adding a fixed k sieve with k being a gmp number.
Your runtime might be dominated though by calculating k % p as that is a very large k. Whatever you do I don't think it is going to be overly fast. [url]http://www.mersenneforum.org/rogue/mtsieve.html[/url][/QUOTE] Thank you for pointing out mtsieve. (I did not know about this before :( ) There is something called modular exponentiation which will calculate k%p very fast. Calculating k%p will take very little time per prime. Addendum: Looks like srsieve is not implemented yet. So this will not help. 
[QUOTE=Citrix;485323]Thank you for pointing out mtsieve. (I did not know about this before :( )
There is something called modular exponentiation which will calculate k%p very fast. Calculating k%p will take very little time per prime. Addendum: Looks like srsieve is not implemented yet. So this will not help.[/QUOTE] There will be a fair bit of manual work although it should be doable in a day or 2. 
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