mersenneforum.org Is there a form of primes similar to primorials but with the sum of primes?
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 2023-02-05, 20:41 #1 hunson   Feb 2020 Germany 23·7 Posts Is there a form of primes similar to primorials but with the sum of primes? I was just wondering if there is a named form of prime numbers which are generated by the formula k*sum(p)+/-1? With sum(p) I mean the sum of all primes up to p, in similar fashion to the primorials k*n#+/-1? I did a google search and did not find an answer, maybe I used the wrong search parameters? Thanks in advance, hunson
2023-02-05, 20:56   #2
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

41×59 Posts

Quote:
 Originally Posted by hunson I was just wondering if there is a named form of prime numbers which are generated by the formula k*sum(p)+/-1? With sum(p) I mean the sum of all primes up to p, in similar fashion to the primorials k*n#+/-1? I did a google search and did not find an answer, maybe I used the wrong search parameters? Thanks in advance, hunson
I think in case of Factorial/Primorial Primes you get very large integers N for which you know all the prime factors for either N+1 or N-1.
With your proposed formula you get much smaller integers N with no easy way of factoring N+1 or N-1.
This would make them not very interesting for large provable primes.
Just my 2 cents.

Last fiddled with by a1call on 2023-02-05 at 20:57

2023-02-05, 21:00   #3
slandrum

Jan 2021
California

2·7·41 Posts

Quote:
 Originally Posted by hunson I was just wondering if there is a named form of prime numbers which are generated by the formula k*sum(p)+/-1? With sum(p) I mean the sum of all primes up to p, in similar fashion to the primorials k*n#+/-1? I did a google search and did not find an answer, maybe I used the wrong search parameters? Thanks in advance, hunson
While I don't know for certain about the answer to your question, I certainly would not expect there to be. There's nothing special about the sum of primes that would suggest anything about the factorability of the sum or numbers near to it other than every other number in the series would be even.

 2023-02-05, 21:05 #4 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 97316 Posts These links might be useful: https://en.wikipedia.org/wiki/Catego..._prime_numbers https://primes.utm.edu/glossary/page...83,(p)%3Dp%2B1. Last fiddled with by a1call on 2023-02-05 at 21:08
2023-02-05, 21:46   #5
chalsall
If I May

"Chris Halsall"
Sep 2002

22×5×571 Posts

Quote:
 Originally Posted by a1call These links might be useful.
Were. Some might find this useful.

Few understand just how deep we can be.

2023-02-05, 22:00   #6
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

41·59 Posts

Quote:
 Originally Posted by chalsall Were. Some might find this useful.
I actually did. Thank you.

2023-02-05, 22:04   #7
chalsall
If I May

"Chris Halsall"
Sep 2002

22·5·571 Posts

Quote:
 Originally Posted by a1call I actually did. Thank you.
No problem. Serious people look out for each other. It's just in our basic nature.

 2023-02-07, 19:00 #8 hunson   Feb 2020 Germany 23×7 Posts Thanks for all the (serious) answers ;) @a1call: You are right, numbers of the form in question do not tent to grow very fast. They are most certainly not suitable for record primes and factoring is much more difficult. Thanks for the links, I will look into it.
2023-02-09, 16:47   #9
Dr Sardonicus

Feb 2017
Nowhere

23·283 Posts

Quote:
 Originally Posted by hunson Thanks for all the (serious) answers ;) @a1call: You are right, numbers of the form in question do not tent to grow very fast. They are most certainly not suitable for record primes and factoring is much more difficult. Thanks for the links, I will look into it.
FWIW, the Prime Number Theorem tells us that

$p_{n}\sim n\log(n)\text{, that is }\frac{n\log(n)}{p_{n}}\rightarrow1\text{ as }n\rightarrow\infty$

so $\sum_{k=1}^n p_{k}$ will be of order n2*log(n).

Asymptotically, the sum divided by n2*log(n) has limit 1/2 as n increases without bound, but for small n the ratio is somewhat larger.

 2023-02-19, 07:24 #10 bur     Aug 2020 79*6581e-4;3*2539e-3 733 Posts What might be more interesting is factoring these numbers. You could even get rid of the +1 and just do the sum of primes. https://oeis.org/A007504 is the respective sequence.
 2023-02-19, 11:37 #11 hunson   Feb 2020 Germany 23×7 Posts Asking as a non mathematician, what is the interesting outcome from factoring the sum of primes or the suggested prime-form? What could be learned from that?

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