20210323, 18:36  #1024  
"Garambois JeanLuc"
Oct 2011
France
13·47 Posts 
Quote:
WOW ! You have found an abundant one for a base which is a prime number ! Congratulations ! This result is wonderful and makes me dizzy ! Unfortunately, I am completely unable to verify this result with the methods at my disposal. The size of the exponent is scary and also the fact that you have to consider the prime numbers < 10^5 instead of 10^4. This slows down the programs very strongly ... I let Happy's C programs run. We will try to find the smallest possible exponent. But I am worried when I see the size of your exponent ! 

20210323, 22:20  #1025  
"Alexander"
Nov 2008
The Alamo City
653 Posts 
Quote:
Quote:


20210324, 07:10  #1026  
Aug 2020
12_{16} Posts 
Quote:
Code:
3^5 * 19^2 * 29^2 * 31 * 37^2 * 47 * 59 * 83 * 103 * 109 * 131 * 139 * 199 * 223 * 271 * 307 * 419 * 523 * 613 * 647 * 691 * 701 * 757 * 811 * 859 * 1063 * 1093 * 1123 * 1151 * 1231 * 1259 * 1381 * 1429 * 1459 * 1481 * 1483 * 1531 * 1559 * 1567 * 1621 * 1783 * 1873 * 1951 * 2269 * 2377 * 2551 * 2707 * 2801 * 2887 * 2971 * 3061 * 3079 * 3083 * 3109 * 3191 * 3257 * 3307 * 3331 * 3631 * 3727 * 3911 * 4003 * 4051 * 4219 * 4591 * 4621 * 4733 * 4931 * 5347 * 5659 * 5743 * 5851 * 6151 * 6217 * 6271 * 6301 * 6661 * 6917 * 6971 * 7177 * 7411 * 7541 * 7591 * 7867 * 8009 * 8101 * 8263 * 8933 * 9103 * 9109 * 9349 * 10099 * 10531 * 10949 * 11287 * 11311 * 11731 * 12211 * 12301 * 12377 * 12433 * 12547 * 13469 * 14009 * 14251 * 14327 * 14653 * 14821 * 14951 * 15121 * 15319 * 15391 * 15541 * 15661 * 15733 * 15913 * 15991 * 16651 * 16763 * 16831 * 17021 * 17443 * 18451 * 18701 * 19609 * 19927 * 20011 * 20021 * 20359 * 20747 * 20749 * 21737 * 22543 * 22621 * 22679 * 23371 * 25117 * 25309 * 26731 * 27551 * 28843 * 29173 * 29251 * 29717 * 31051 * 31081 * 32063 * 32191 * 33151 * 33211 * 33301 * 33967 * 34273 * 35531 * 35671 * 35729 * 35803 * 36671 * 38611 * 39443 * 40591 * 41413 * 42979 * 43291 * 44371 * 46171 * 46399 * 47881 * 48907 * 50923 * 51679 * 51949 * 52027 * 52837 * 52973 * 54367 * 56167 * 56701 * 58631 * 59053 * 59509 * 59671 * 60089 * 60589 * 60611 * 60763 * 62701 * 64091 * 65269 * 65437 * 65551 * 67489 * 67651 * 68311 * 69191 * 69499 * 70111 * 70841 * 71341 * 71707 * 72931 * 73951 * 75401 * 76039 * 76561 * 77141 * 77419 * 78541 * 78737 * 80191 * 81001 * 81901 * 81919 * 84391 * 84457 * 84871 * 86131 * 87211 * 92251 * 94351 * 94771 * 95701 * 97021 * 98533 * 99877 I am not quite sure with my coding skill. I would really appreciate if someone can double check it. 

20210324, 13:26  #1027  
"Alexander"
Nov 2008
The Alamo City
28D_{16} Posts 
Quote:


20210324, 14:24  #1028 
"Oliver"
Sep 2017
Porta Westfalica, DE
23^{2} Posts 
We can do that more efficiently. Since we have \[\sigma(p^n)=\sum^{n1}_{i=0}{p^i}=\frac{p^n1}{p1}\] for prime \(p\), we can check for divisors of \(\sigma(p)\) as such: compute \(p^n \mod (p  1) \cdot f\), where \(f\) is the factor to be checked. If the result is 1, \(f\) divides \(\sigma(p^n)\). Since modular exponentiation is cheap, we can do it extremely fast.
Last fiddled with by kruoli on 20210324 at 14:28 Reason: Removed bogus letters. Corrected formula. 
20210324, 15:43  #1029  
"Alexander"
Nov 2008
The Alamo City
653 Posts 
Quote:


20210324, 19:29  #1030 
"Garambois JeanLuc"
Oct 2011
France
13×47 Posts 
Thank you very much to you for this validation work !
I will examine all of this carefully so that I too can learn how to do this ... 
20210324, 19:37  #1031  
"Alexander"
Nov 2008
The Alamo City
28D_{16} Posts 
Quote:
Quote:
Edit: 37^2 does divide, according to the new code, so we're OK there. Last fiddled with by Happy5214 on 20210324 at 19:42 

20210324, 19:53  #1032  
Aug 2020
2·3^{2} Posts 
Quote:


20210324, 21:08  #1033  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{2}×13×113 Posts 
Quote:


20210324, 21:19  #1034 
"Alexander"
Nov 2008
The Alamo City
653 Posts 
Given the name, audience, use case, and lack of actual values to use it on, I figured it was unnecessary. I guess not. I'll add it (which is, what, 4 lines?) when I get back to my main computer tomorrow. Until then, do not pass it a composite base.
Last fiddled with by Happy5214 on 20210324 at 21:20 
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