20220705, 12:20  #1 
Jul 2022
11_{2} Posts 
Primes of the form 'perfect number' + 1 (a.k.a. RPN)
Let P_{n} denote the nth even perfect number.
Related to Mersenne primes, currently 51 perfect numbers are known. Primes of the form are also listed in OEIS. Currently 4 primes are known: Are all other cases up to composite? Is a factor known of ? I found no factor below . 
20220705, 13:57  #2 
Mar 2019
123_{16} Posts 
You can test this yourself with pfgw. Prepare a file like:
Code:
ABC 2^($a1)*(2^$a1)+1 521 607 1279 2203 2281 3217 4253 4423 9689 9941 11213 19937 21701 23209 44497 86243 110503 132049 216091 ... Code:
./pfgw64 f10 lpfgw.out T8 ./nearperf.txt Code:
Recognized ABC Sieve file: 2^(5211)*(2^5211)+1 has factors: 7 2^(6071)*(2^6071)+1 has factors: 11 2^(12791)*(2^12791)+1 is composite: RES64: [570A6B3FD91E6339] (0.0154s+0.0010s) 2^(22031)*(2^22031)+1 is composite: RES64: [ECB4FE924C674723] (0.0244s+0.0010s) 2^(22811)*(2^22811)+1 has factors: 197 2^(32171)*(2^32171)+1 has factors: 11 2^(42531)*(2^42531)+1 has factors: 7 2^(44231)*(2^44231)+1 is composite: RES64: [F3603EEF4BD4F197] (0.0471s+0.0016s) 2^(96891)*(2^96891)+1 has factors: 7 2^(99411)*(2^99411)+1 has factors: 7 2^(112131)*(2^112131)+1 has factors: 7 2^(199371)*(2^199371)+1 has factors: 7 2^(217011)*(2^217011)+1 has factors: 7 2^(232091)*(2^232091)+1 has factors: 35603 2^(444971)*(2^444971)+1 has factors: 11 2^(862431)*(2^862431)+1 has factors: 7 2^(1105031)*(2^1105031)+1 has factors: 491 2^(1320491)*(2^1320491)+1 is composite: RES64: [1B3B60AEC3578817] (45.9414s+0.4677s) 2^(2160911)*(2^2160911)+1 has factors: 4673 2^(7568391)*(2^7568391)+1 has factors: 7 2^(8594331)*(2^8594331)+1 has factors: 7 2^(12577871)*(2^12577871)+1 has factors: 11 2^(13982691)*(2^13982691)+1 has factors: 7 2^(29762211)*(2^29762211)+1 has factors: 7 2^(30213771)*(2^30213771)+1 has factors: 7 2^(69725931)*(2^69725931)+1 has factors: 7 2^(134669171)*(2^134669171)+1 has factors: 11 2^(209960111)*(2^209960111)+1 has factors: 1552147 2^(240365831)*(2^240365831)+1 has factors: 149 2^(259649511)*(2^259649511)+1 has factors: 7 2^(304024571)*(2^304024571)+1 has factors: 11^2 2^(325826571)*(2^325826571)+1 has factors: 7 2^(371566671)*(2^371566671)+1 has factors: 7 2^(426438011)*(2^426438011)+1 has factors: 3593 2^(431126091)*(2^431126091)+1 has factors: 7 2^(578851611)*(2^578851611)+1 has factors: 7 2^(601091871)*(2^601091871)+1 has factors: 7 
20220705, 14:05  #3 
Einyen
Dec 2003
Denmark
3,347 Posts 
We have an old thread for this:
https://www.mersenneforum.org/showthread.php?t=10336 The list is in post #10, we are only missing factor for that one for p=74207281. I checked to 6*10^{12} back then when that Mersenne Prime was found. 
20220705, 14:10  #4 
Aug 2020
79*6581e4;3*2539e3
5^{2}·23 Posts 
These should have very low weight due to k being a very large prime.
Did you test for factors or have factors of 7 < n < 49? LLR2 didn't like even the 8th one: Code:
$ ./sllr2 d q"2147483647*2^30+1" 2147483647 > 2^30, so we can only do a PRP test for 2147483647*2^30+1. Iter: 1/31, ERROR: ILLEGAL SUMOUT Possible hardware failure, consult the readme file. Continuing from last save file. Waiting five minutes before restarting. 
20220705, 14:36  #5 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3,559 Posts 
Can “perfect number  3” (except 3) be prime?

20220705, 14:56  #6 
Apr 2020
793_{10} Posts 

20220705, 20:46  #7 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6558_{10} Posts 

20220705, 22:15  #8 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}·1,667 Posts 
First LL by "b3g" https://www.mersenne.org/report_expo...exp_hi=&full=1 which stands for "beyond third generation"
Last fiddled with by kriesel on 20220705 at 22:19 
20220705, 23:43  #9  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9901_{10} Posts 
Quote:
/sigh/ There is so much excitement doing something that was done to death years ago, right? Much more excitement than to read the link that was helpfully provided. That thread has everything you ever wanted to know. Read it! 

20220705, 23:56  #10 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}·1,667 Posts 
A current link, as a substitute for the dead link for the sequence in https://www.mersenneforum.org/showpo...61&postcount=3 is https://oeis.org/search?q=a061644&la...lish&go=Search
or more concisely, https://oeis.org/A061644 edit: which I see has now been updated in place, silently (without note by whichever mod edited that 14 year old post this evening). Old link shown with resulting error response below. Apparently a valid link when originally posted, from before the transition from AT&T to OEIS.org in 20092010. https://oeis.org/wiki/Welcome#OEIS:_Brief_History Last fiddled with by kriesel on 20220706 at 00:29 
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