20190505, 21:14  #1  
Apr 2019
5·41 Posts 
Observations of Wieferich primes and Wieferich1 friendly club
This is a discussion on the forms of numbers found here: http://oeis.org/A239875
Quote:
It is noted on that page that numbers of the form (39 * perfect numbers) also follow this abundancy index. One thing to test is the primality of these (39 * perfect numbers)+1: (39*(2^(31))*((2^3)1))+1 = 1093 (one of the known Wieferich primes) (39*(2^(311))*((2^31)1))+1 = 89927877317458132993 (prime but not Wieferich prime) I am attempting to test the remaining perfect numbers based on currently known mersenne primes, using PFGW: Code:
ABC (39*(2^($a1))*((2^$a)1))+1 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423 9689 9941 11213 19937 21701 23209 44497 86243 110503 132049 216091 756839 859433 1257787 1398269 2976221 3021377 6972593 13466917 20996011 24036583 25964951 30402457 32582657 37156667 42643801 43112609 57885161 74207281 77232917 82589933 Has this been checked before? One other observation/conjecture I have made, is regarding the prime factors of (Wieferich primes1) and friends. It seems that they are composed entirely of mersenne prime exponents, or mersenne primes themselves. This is trivially true for 39*perfect number form, but also holds for the two known numbers in the sequence which do not follow that form: a(2)=3510 = 2^1 * 3^3 * 5^1 * 13^1 (2,3,5,13 are Mp exponents) a(5)=11504844 = 2^2 * 3^2 * 13^2 * 31^1 * 61^1 In this case the factor 31 could be viewed as either the exponent of M31, or M5 itself. Similarly the factor 3 could be considered an exponent of M3, or M2 itself. So with this taken into account, it may be that a mersenne prime factor always occurs exactly once for these numbers, with the remaining factors being entirely mersenne prime exponents. Any ideas if some form of this conjecture could be proven? I haven't seen any mention of such an idea before, but maybe its been thought of already, or that it follows somewhat easily from some related theorems? Another note from the oeis page on Wieferich primes themselves: http://oeis.org/A001220 Quote:
Code:
Base 2 conversions: 1092 = 10001000100 (palindrome * 2^2) 3510 = 110110110110 (palindrome * 2^1) 19344 = 100101110010000 (not pseudopalindrome) 316992 = 1001101011001000000 (palindrome * 2^6) 11504844 = 101011111000110011001100 (not pseudopalindrome) 1308463104 = 1001101111111011001000000000000 (palindrome * 2^12) 335004893184 = 100110111111111110110010000000000000000 = (palindrome * 2^16) 5360108961792 = 1001101111111111111011001000000000000000000 = (palindrome * 2^18) 89927877317458132992 = 1001101111111111111111111111111011001000000000000000000000000000000 = (palindrome * 2^30) 103679783671223438041533012022199844864 = 1001101111111111111111111111111111111111111111111111111111111011001000000000000000000000000000000000000000000000000000000000000 (palindrome * 2^60) 

20190505, 21:45  #2 
Sep 2003
3^{2}·7·41 Posts 
I think maybe the Wieferich@Home project tested repeating bit patterns to try to find a new Wieferich prime. Or maybe they just exhaustively tested all values, I don't know.
They didn't find one, and now that project is defunct and the web domain has been taken over by someone else for a different purpose. Here is an archived stats page from 2016 from the Internet archive. Have you checked if the odd coincidences (bit patterns and friendly patterns) of the two Wieferich primes are also applicable in some way to the Basea Wieferich primes for bases other than 2? PS, needless to say, if any factor of a Mersenne number with prime exponent turns out to be nonsquarefree, it would automatically become a new Wieferich prime. I regularly check newly discovered factors, but obviously the odds are astronomical. The nonbase2 Wieferich primes do have some cases that correspond to nonsquarefree factors. For example: Code:
3^5−1 = 11^2 * ... 7^2+1 = 5^2 * ... 13^431+1 = 863^2 * ... 17^24473−1 = 48947^2 * ... 18^2+1 = 5^2 * ... 18^3−1 = 7^3 * ... 18^214007+1 = 1284043^2 * ... 19^3+1 = 7^3 * ... 22^3−1 = 13^2 * ... 22^246183293−1 = 492366587^2 * ... 22^44999368331−1 = 9809862296159^2 * ... 24^2−1 = 5^2 * ... 26^2−1 = 5^2 * ... 26^7+1 = 71^2 * ... 26^3347628353+1 = 6695256707^2 * ... 28^11+1 = 23^2 * ... 30^3−1 = 7^2 * ... (2^14)^13 + 1 is divided by 1093^2 (2^26)^7 + 1 is divided by 1093^2 (2^28)^13 − 1 is divided by 1093^2 (2^135)^13 − 1 is divided by 3511^2 (2^270)^13 − 1 is divided by 3511^2 (2^351)^5 − 1 is divided by 3511^2 But of course, it's no big deal if a Mersenne number with composite exponent has a nonsquarefree factor. 
20190511, 18:26  #3  
Apr 2019
5·41 Posts 
Quote:
I did stumble upon the definition of khyperperfect numbers, which seems to be a reasonable generalization of such a thing (where k=base1). For base 3 as an example, there are 2hyperperfect numbers: https://oeis.org/A007593 Quote:
However this lacks the property that all primes of this form have a prime exponent. Another generalization of mersenne numbers(primes) could be: (b^n1)/(b1) (AKA repunits in base b) This actually does seem to have the property where only prime exponents produce primes. So mersenne numbers having base 2 are unique in this case because its the only base where these two generalized forms are actually equivalent: 2^n(21) == (2^n1)/(21) It may be that (if there is any relevance to the patterns in my OP at all), that they only apply to base 2 wieferich's. One last thing which I tried to understand, but got a bit stuck, is if there is any way to define a "khyperabundancy index", or "khyperfriendlyness" Such that all khyperperfect numbers are khyperfriendly for a given k, and also where this generalized formula is equivalent to the regular abundancy index given k=1. Edit: I should also note, that for the case of "pseudopalindromes", I did not see any such patterns for wieferich primes of other prime bases, (base 3, or base 5 for example). Last fiddled with by hansl on 20190511 at 18:40 Reason: palindrome note 

20200902, 10:40  #4 
"Jeppe"
Jan 2016
Denmark
2^{2}×41 Posts 
I wrote that OEIS entry. Did you ever complete that ABC file to 2976221 or beyond? /JeppeSN

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