Is there a form of primes similar to primorials but with the sum of primes?

hunson · 2023-02-05, 20:41

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

a1call · 2023-02-05, 20:56

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.

slandrum · 2023-02-05, 21:00

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.

a1call · 2023-02-05, 21:05

These links might be useful:
https://en.wikipedia.org/wiki/Catego..._prime_numbers
https://primes.utm.edu/glossary/page...83,(p)%3Dp%2B1.

chalsall · 2023-02-05, 21:46

Quote:

Originally Posted by a1call

These links might be useful.

Were. Some might find this useful.

Few understand just how deep we can be.

a1call · 2023-02-05, 22:00

Quote:

Originally Posted by chalsall

Were. Some might find this useful.

I actually did. Thank you.

chalsall · 2023-02-05, 22:04

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.

hunson · 2023-02-07, 19:00

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.

Dr Sardonicus · 2023-02-09, 16:47

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.

bur · 2023-02-19, 07:24

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.

hunson · 2023-02-19, 11:37

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?

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 ksum(p)+/-1? With sum(p) I mean the sum of all primes up to p, in similar fashion to the primorials kn#+/-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, 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

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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.

2023-02-19, 07:24	#10
bur Aug 2020 796581e-4;32539e-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?