[seqfan] Re: A look at prime number gaps using only potential prime numbers

Harry Neel neelh48 at hotmail.com
Sun Oct 3 20:07:53 CEST 2021

Thanks for your input and suggestions.  I will need to think about how to use some examples without creating a lot of confusion.

Your reply contains the statement "(Top-posting, since Harry seems to prefer that.)"  What does this mean?

As I think of a potentially different way to search for related information I have attempted to do so.

I do not recall if I have checked A005250 and A0022386. I will do so, of course.


Sent from Outlook<http://aka.ms/weboutlook>

Harry E. Neel​

From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of hv at crypt.org <hv at crypt.org>
Sent: Sunday, October 3, 2021 12:41 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: A look at prime number gaps using only potential prime numbers

(Top-posting, since Harry seems to prefer that.)

Sorry, the attachments didn't come through: I believe the list does not
permit them.

It would probably make things clearer if you could refer to existing
sequences in the OEIS when relevant. For example, when you refer to
"Maximal Prime Numbers Gaps" I think you may mean A005250, but I'm not
certain. (A002386 may also be relevant.)

In any case, I doubt there's much more I can contribute here.


Harry Neel <neelh48 at hotmail.com> wrote:
:Thank you for your response and I am grateful for your verifications.  The script may be 'simple' but I do not have the tools myself.
:I have attempted to answer your questions. Please correct where I make incorrect wording or expressions.
:I am taking a liberty to attach two PDF documents. One is a sieve form (of which there are many) of the first part of the sieve I mentioned in my inquiry.  Not all patterns are identified, but hopefully it will help.
:The second is a comparison between the gaps in the sieve to the Maximal Prime Numbers Gaps.
:I used the larger Maximal Prime Gap figures as a comparison to the potential prime location gaps because that is what I had. I did not know if the values I used were maximal values, or not.  They were not, and I am not surprized. Thank you for showing that the potiential prime maximal gaps of larger primes are indeed different.
:I did not bother even mentioning the MERITs of these gaps, one because I have no idea if they would be part of a sequence (or how), and two because there was no way to determine if there was any interest unless the question was asked.
:Sent from Outlook<http://aka.ms/weboutlook>
:Harry E. Neel​
:From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of hv at crypt.org <hv at crypt.org>
:Sent: Monday, September 27, 2021 11:56 PM
:To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
:Subject: [seqfan] Re: A look at prime number gaps using only potential prime numbers
:Harry Neel <neelh48 at hotmail.com> wrote:
::Looking at a sieve for all numbers having factors greater than 5 and
::selecting choosing numbers that have factors equal to greater than 7
::it appears that a lists of 'Maximal Gaps' that is based on where
::potential prime locations are not prime numbers.  For example 49 is
::the first potential prime location that is not prime having a prime
::factor of 7.
::The preceding prime is 47 and the next number which could be prime is
::53, which is prime. So is there a maximal gap in the potential prime
::locations of 1?
:If I understand this correctly, between any pair of consecutive primes
:you're counting how many of the intervening composites are coprime with
:each of 2, 3 and 5.
:This may be the result. I just took a sieve containing the numbers greater than 10 with a prime factor greater than or equal to 7. Then, assuming they are all 'potential prime numbers' before factoring began. The first gap is at potential prime location 49, as 47 before and 53 are prime then the gap in the potential prime locations is 1.
::Unless mistakes were made, the gaps below are straight from sieving.
::Gap of                 following prime
::1                                               47
::2                                             113
::3                                              317
::4                                              523
::8                                              1327
::9                                              9551
::10                                           15683
:On the basis of my understanding above I can confirm those, and offer
:some more using the simple perl program below:
:gap, prime, next prime
:1 47 53
:2 113 127
:3 317 331
:4 523 541
:8 1327 1361
:9 9551 9587
:10 15683 15727
:12 19609 19661
:13 25471 25523
:18 31397 31469
:22 155921 156007
:24 360653 360749
:29 370261 370373
:30 1349533 1349651
:34 1357201 1357333
:38 2010733 2010881
:40 4652353 4652507
:47 17051707 17051887
:55 20831323 20831533
:58 47326693 47326913
:59 122164747 122164969
:61 189695659 189695893
:65 191912783 191913031
:66 387096133 387096383
:74 436273009 436273291
:76 1294268491 1294268779
:77 1453168141 1453168433
:85 2300942549 2300942869
::Only preliminary checks on additional gaps have been preformed and
::Maximal Gap lengths may not be appropriate.  Gaps is the potential
::prime locations for compared to Maximal Primes appear to be:
::Gap in                                   following             Maximal               *If using a gap of 1 between 2 or 3
::Potential Prime                 prime                    prime gap*         use higher value.
::11                                           360653                  95 or 96
::12                                           370261                  111 or 112
::13                                           1349533                117 or118
::And a few more up through
::67                                           191912783           247 or 248
:Sorry, I don't understand what this table represents.
:Gap in Potential Prime Locations is the size of the gap in the sieve.
:Following Prime is the prime number preceeding the gap in the sieve.
:The Maximal Prime Gap column is the established gap size that includes every composite number within the gap.
:(Some use the p2-p1 as the gap size. Some use (p2-p1)-1 as the gaps size.  Therefor the gap between the primes 2 and 3 is either 0, or 1.)
:It was included only the provided information about the difference between the two gap sizes.
::Is there any validity here?  Worth examination?  Help from someone
::will definitely be needed as I have gone about as far as reasonable
::by doing everything by hand and spreadsheet.
:As a sequence, my main question would be what motivates sieving out
:multiples of 2, 3 and 5 - why not just 2 and 3, or all of 2, 3, 5 and 7?
:I am not sure that I can answer your question.  What I can say is that when I searched OEIS for the sequence 47,113,317,523,1327 I received a "terms do not match" message.
:As these are the prime numbers preceeding 'maximal gaps' in the sieve for potential primes locations (numbers) for all numbers having prime factors greater than or equal to 7 I was surprized.
:There is a certain symetry to the sieve that is appealing as it can allow a visiual for some aspects concerning primes. For example, the maximum number of primes that are 30 units apart is 6. This is simply because the prime factor 7 will always prevent a larger sequence.  The groups of 6 primes 30 units appart is one of the hosts of sequences related to primes.
:I have been looking at this for well over a year now, maybe longer. Finally decided to ask if there is any validity in the sequence, or not.
::I modified a portion of the sieve I use if anyone is interested.  Also
::have a preliminary comparison between using gaps in the sieve of
::potential prime locations and associated Maximal Prime Gaps.
::Thanks for your attention.
::H. Neel
:Hugo van der Sanden
:use strict;
:use warnings;
:use Math::Prime::Util qw{ next_prime gcd };
:my $p = 2;
:my $best = 0;
:my @coprime = map { (gcd($_, 30) == 1) ? 1 : 0 } 0 .. 29;
:while (1) {
:    my $np = next_prime($p);
:    my $gap = 0;
:    for ($p + 1 .. $np - 1) {
:        $gap += $coprime[$_ % 30];
:    }
:    if ($best < $gap) {
:        print "$gap $p $np\n";
:        $best = $gap;
:    }
:    $p = $np;
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