Primes in A048788 OEIS Sequence

carpetpool · 2017-03-16, 04:08

Is anyone able to check which indices a(n) A048788 OEIS sequence is prime for for n up to 20k or more? How can PFGW define recurrence relation non-Lucas Sequences so I know how to test them? Manual primality tests yield that a(n) is prime for the following n (up to 317):

2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317

OEIS doesn't have a sequence for this, so is someone also able to verify the current results on the indices n such that a(n) is prime for. Thanks for help and feedback. More work and time also appreciated.

science_man_88 · 2017-03-16, 11:07

Quote:

Originally Posted by carpetpool

Is anyone able to check which indices a(n) A048788 OEIS sequence is prime for for n up to 20k or more? How can PFGW define recurrence relation non-Lucas Sequences so I know how to test them? Manual primality tests yield that a(n) is prime for the following n (up to 317):

2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317

OEIS doesn't have a sequence for this, so is someone also able to verify the current results on the indices n such that a(n) is prime for. Thanks for help and feedback. More work and time also appreciated.

by the rule given in june 2014 all indices of primes will be in prime locations. in the case of y=2n+1; a(y)=3*a(y-2)-(-1*a(y-3))

CRGreathouse · 2017-03-16, 20:42

First, note that a(n) > 1 for n > 1 and that the sequence is a strong divisibility sequence, so if a(n) is prime then n is prime. So we need only check prime indices.

I added a closed-form program for the sequence to make it less of a pain to work with and started crunching away. The first minute or so gave these terms:
2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317, 491, 647, 719, 947, 1039, 1193, 1499, 1867, 1933, 4591, 7127

CRGreathouse · 2017-03-16, 20:43

Quote:

Originally Posted by science_man_88

by the rule given in june 2014 all indices of primes will be in prime locations.

Well-spotted, you beat me to it.

CRGreathouse · 2017-03-16, 21:12

A few more: 11273, 13591, 16823, 18541

science_man_88 · 2017-03-16, 21:27

Quote:

Originally Posted by science_man_88

by the rule given in june 2014 all indices of primes will be in prime locations. in the case of y=2n+1; a(y)=3*a(y-2)-(-1*a(y-3))

of course all this gets to the fact that a(y-3) is forcing which types of primes are most popular in the sequence. and since it's an strongly increasing sequence once certain values are missed they aren't going to show up ever. I haven't thought about how fast it's increasing and what that says about which may be prime or not. anyways I may post again when I get more in my head. other sequences that may help you Edit:

https://oeis.org/A001835 all the odd placed terms.
https://oeis.org/A001353 twice this one is every second of the sequence you want to talk about and these fall into a lucas sequence.

carpetpool · 2017-03-16, 23:50

Thanks, CRGreathouse for those terms. To make sure (so an OEIS sequence can be added), are these the correct indices n (for a(n)) in order up to n = 20k?

I will try to add in an OEIS sequence if so.

Also just for curiosity, what is the density for the prime indices n such that a(n) is prime for compared to the primes n such that 2^n-1 (Mersenne) is prime for?

In other words, for any prime n, what is the (approximate) probability that a(n) is prime for, and for any given prime n, is there a greater chance that a(n) or 2^n-1 is prime for?

CRGreathouse · 2017-03-17, 01:32

Quote:

Originally Posted by carpetpool

Thanks, CRGreathouse for those terms. To make sure (so an OEIS sequence can be added), are these the correct indices n (for a(n)) in order up to n = 20k?

I wasn't making that claim, but it happens to be true as the next two are 27947 and 34351.

Quote:

Originally Posted by carpetpool

Also just for curiosity, what is the density for the prime indices n such that a(n) is prime for compared to the primes n such that 2^n-1 (Mersenne) is prime for?

In other words, for any prime n, what is the (approximate) probability that a(n) is prime for, and for any given prime n, is there a greater chance that a(n) or 2^n-1 is prime for?

I don't know. This sequence grows slower than 2^n-1, with its main term being 1.9318516...^n, but it really comes down to its behavior on small prime factors.

carpetpool · 2017-03-17, 12:22

So I see you are continuing to test this sequence, or not?

I would be happy to check these terms too, though unfortunately I am stuck on defining a double-recurrence relation with PFGW like sm88 this sequence a(n) was. If all n <= 1M were tested, these terms would be checked to 300k digits approximately? Not saying anyone should risk their computer to do all that work. (I certainly would only go to about n = 200k or 300k) which is about 60k digits. I am only making sure I got the approximations of how large these primes are.

CRGreathouse · 2017-03-17, 22:57

OK, terms are

2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317, 491, 647, 719, 947, 1039, 1193, 1499, 1867, 1933, 4591, 7127, 11273, 13591, 16823, 18541, 27947, 34351, 66841, 80051, 80629, 81547

with no others through 86743. You can take it from here if you like, I'm done. The last term corresponds to a PRP23319. Through a million would be 286k digits.

2017-03-16, 04:08	#1
carpetpool "Sam" Nov 2016 5·67 Posts	Primes in A048788 OEIS Sequence Is anyone able to check which indices a(n) A048788 OEIS sequence is prime for for n up to 20k or more? How can PFGW define recurrence relation non-Lucas Sequences so I know how to test them? Manual primality tests yield that a(n) is prime for the following n (up to 317): 2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317 OEIS doesn't have a sequence for this, so is someone also able to verify the current results on the indices n such that a(n) is prime for. Thanks for help and feedback. More work and time also appreciated.

2017-03-16, 23:50	#7
carpetpool "Sam" Nov 2016 101001111₂ Posts	Thanks, CRGreathouse for those terms. To make sure (so an OEIS sequence can be added), are these the correct indices n (for a(n)) in order up to n = 20k? I will try to add in an OEIS sequence if so. Also just for curiosity, what is the density for the prime indices n such that a(n) is prime for compared to the primes n such that 2^n-1 (Mersenne) is prime for? In other words, for any prime n, what is the (approximate) probability that a(n) is prime for, and for any given prime n, is there a greater chance that a(n) or 2^n-1 is prime for? Last fiddled with by carpetpool on 2017-03-16 at 23:56

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2017-03-16, 20:42	#3
CRGreathouse Aug 2006 2²·3·499 Posts	First, note that a(n) > 1 for n > 1 and that the sequence is a strong divisibility sequence, so if a(n) is prime then n is prime. So we need only check prime indices. I added a closed-form program for the sequence to make it less of a pain to work with and started crunching away. The first minute or so gave these terms: 2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317, 491, 647, 719, 947, 1039, 1193, 1499, 1867, 1933, 4591, 7127

2017-03-16, 21:12	#5
CRGreathouse Aug 2006 2²×3×499 Posts	A few more: 11273, 13591, 16823, 18541

2017-03-17, 12:22	#9
carpetpool "Sam" Nov 2016 5×67 Posts	So I see you are continuing to test this sequence, or not? I would be happy to check these terms too, though unfortunately I am stuck on defining a double-recurrence relation with PFGW like sm88 this sequence a(n) was. If all n <= 1M were tested, these terms would be checked to 300k digits approximately? Not saying anyone should risk their computer to do all that work. (I certainly would only go to about n = 200k or 300k) which is about 60k digits. I am only making sure I got the approximations of how large these primes are.

2017-03-17, 22:57	#10
CRGreathouse Aug 2006 2²×3×499 Posts	OK, terms are 2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 149, 227, 277, 311, 317, 491, 647, 719, 947, 1039, 1193, 1499, 1867, 1933, 4591, 7127, 11273, 13591, 16823, 18541, 27947, 34351, 66841, 80051, 80629, 81547 with no others through 86743. You can take it from here if you like, I'm done. The last term corresponds to a PRP23319. Through a million would be 286k digits.