mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Blogorrhea > enzocreti

Reply
 
Thread Tools
Old 2019-04-13, 21:43   #1
enzocreti
 
Mar 2018

2·3·5·17 Posts
Default I submitted a new sequence in Oeis

Primes p such p+2 has two distinct prime factors whose at least one of them is raised to a power greater than 1.

DATA 43, 61, 73, 97, 151, 173, 223, 277, 313, 331, 349, 367, 373, 421, 439, 457, 523, 547, 601, 619, 673, 691, 709, 733, 773, 823, 853, 907, 929, 997, 1033, 1051, 1069, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1303, 1321, 1373, 1423, 1429, 1447




Maybe they will pubblish in Oeis...a question: why all the terms are of the form 6m+1? The terms from 61 to 277 are sexy primes.


Anyway there is some mistake...313 shoudnt be in the sequence because 315 has three and not two distinc prime factors...the same for 1033

Last fiddled with by enzocreti on 2019-04-13 at 21:57
enzocreti is offline   Reply With Quote
Old 2019-04-14, 01:11   #2
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

63718 Posts
Default

Quote:
Originally Posted by enzocreti View Post
Primes p such p+2 has two distinct prime factors whose at least one of them is raised to a power greater than 1.

DATA 43, 61, 73, 97, 151, 173, 223, 277, 313, 331, 349, 367, 373, 421, 439, 457, 523, 547, 601, 619, 673, 691, 709, 733, 773, 823, 853, 907, 929, 997, 1033, 1051, 1069, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1303, 1321, 1373, 1423, 1429, 1447




Maybe they will pubblish in Oeis...a question: why all the terms are of the form 6m+1? The terms from 61 to 277 are sexy primes.


Anyway there is some mistake...313 shoudnt be in the sequence because 315 has three and not two distinc prime factors...the same for 1033
In fact, p = 313, 523, 691, 733, 823, 853, 1033, 1069, 1303, 1423, and 1447 are all such that p+2 has 3 prime factors.

Up to the limit 2000, I get the following list:

[43, 61, 73, 97, 151, 173, 223, 277, 331, 349, 367, 373, 421, 439, 457, 547, 601, 619, 673, 709, 773, 907, 929, 997, 1051, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1321, 1373, 1429, 1523, 1571, 1609, 1627, 1699, 1789, 1811, 1823, 1861, 1873, 1973].

The prime p = 173 is the first counterexample to your implicit assertion that "all the terms are of the form 6m+1."
Dr Sardonicus is offline   Reply With Quote
Old 2019-04-14, 14:23   #3
enzocreti
 
Mar 2018

51010 Posts
Default Another oeis sequence submitted

13, 19, 31, 37, 43, 53, 61, 67, 73, 83, 89, 97, 109, 113, 127, 131, 139, 151, 157, 173, 181, 199, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409, 421, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541


primes p such p+2 has exactly two distinct prime factors


first of all seem to exist infinitely many emirps pairs like (13,31) (37,73)....

and there seems to be infinitely many triplets like (61,67,73) , (251,257,263) (367,373,379) which differ by 6

See also that 61,251,367 the first terms of the triplets belong to Oeis sequence A038107 number of primes<n^2

Last fiddled with by enzocreti on 2019-04-14 at 14:34
enzocreti is offline   Reply With Quote
Old 2019-04-18, 02:12   #4
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

10110111001112 Posts
Default

Quote:
Originally Posted by Dr Sardonicus View Post
Up to the limit 2000, I get the following list:

[43, 61, 73, 97, 151, 173, 223, 277, 331, 349, 367, 373, 421, 439, 457, 547, 601, 619, 673, 709, 773, 907, 929, 997, 1051, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1321, 1373, 1429, 1523, 1571, 1609, 1627, 1699, 1789, 1811, 1823, 1861, 1873, 1973].
I concur, and find 1,920,518 such numbers up to a billion. PARI/GP:
Code:
v=List();forfactored(n=2,10^9+2,f=n[2];e=f[,2]; if(#e==2 && vecmax(e)>1 && isprime(n[1]-2), listput(v,n[1]-2))); #v
Quote:
Originally Posted by Dr Sardonicus View Post
The prime p = 173 is the first counterexample to your implicit assertion that "all the terms are of the form 6m+1."
Up to a billion 1,335,006 (69.5%) are 1 mod 6 and 585,512 (30.5%) are 5 mod 6.
CRGreathouse is offline   Reply With Quote
Old 2019-04-18, 15:06   #5
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

332110 Posts
Default

Quote:
Originally Posted by enzocreti View Post
Primes p such p+2 has two distinct prime factors whose at least one of them is raised to a power greater than 1.
<snip>
a question: why all the terms are of the form 6m+1?
<snip>
It occurred to me to look at the congruence classes of p (mod 18). The result was a triumph for elementary number theory.

If p = 18*k + 1 then 3 divides p + 2, and (p+2)/3 = 6*k + 1. In order to satisfy the conditions, this has to be the square or higher power of a prime.

If p = 18*k + 5, then p + 2 is not divisible by 2 or 3.

If p == 18*k + 7 then p+2 is divisible by 9. If e = valuation(p+2, 3) then p satisfies the condition if (p+2)/3^e is prime, or a square or higher power of a prime.

If p = 18*k + 11 then p + 2 is not divisible by 2 or 3.

If p = 18*k + 13, then (p + 2)/3 = 6*k+ 5. This cannot be a square, so the only way p can satisfy the condition is for (p+2)/3 to be the cube or higher (odd) power of a prime.

If p = 18*k + 17, then p + 2 is not divisible by 2 or 3.

Accordingly, p = 18*k + 7 should occur most often; the built-in square factor 9 of p+2 gives this congruence class a strong advantage. The classes p = 18*k + 5, 18*k + 11, and 18*k + 17 should, under the "assumption of ignorance" occur about equally often, but (due to the lack of any known square factor) probably less often than 18*k + 7. The p = 18*k + 1 satisfying the conditions should be quite rare, and p = 18*k + 13 rarest of all.

Following is a count of the number of p satisfying the conditions up to the limit 50 million, followed by a vector with the breakdown into the residue classes 1, 5, 7, 11, 13, 17 (mod 18). After that, is the list of the six primes p = 18*k + 13 less than 50 million which satisfy the conditions.

140173 [104, 14562, 96666, 14437, 6, 14398]

p = 373, 14737, 6744271, 15533149, 19309027, 22936117

Last fiddled with by Dr Sardonicus on 2019-04-18 at 15:11 Reason: <snip>
Dr Sardonicus is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Low range P-1, should NF be submitted? M344587487 PrimeNet 11 2019-04-20 06:49
OEIS sequence A088782 sweety439 And now for something completely different 7 2018-04-30 22:52
OEIS sequence A027750 MattcAnderson MattcAnderson 2 2017-04-26 22:19
Primes in A048788 OEIS Sequence carpetpool Miscellaneous Math 9 2017-03-17 22:57
Have I submitted bad data? (Or, well, about to anyway) Nazo Software 5 2005-08-06 05:44

All times are UTC. The time now is 21:26.

Fri Aug 7 21:26:50 UTC 2020 up 21 days, 17:13, 1 user, load averages: 2.41, 2.22, 2.14

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.