20180918, 16:06  #1 
May 2018
53_{8} Posts 
Numbers of the form 1!+2!+3!+...
For no special reasons I started getting interested in factoring numbers of the form or . Do you know if someone has already looked into these numbers? Of course a lot of them are done on factodb, and some are quite easy with ECM, but sometimes the factorization is not so easy.
Last fiddled with by ricky on 20180918 at 16:07 
20180918, 16:35  #2 
Aug 2006
3·5^{2}·79 Posts 
For one thing, Zivkovic proved that there are only finitely many primes of the latter form. You may find more information on their OEIS entries:
https://oeis.org/A007489 https://oeis.org/A003422 
20180918, 16:58  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×13×73 Posts 
That is
1+2+6+24+120.... So except for the first term it is always divisible by 3 and except for the 2nd term it is never prime. After the 6th term it will always be divisible by 3 only once and the same type of progression will apply to infinity. 
20180918, 22:53  #4 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·13·73 Posts 
As usual spoke before checking first.
Apparently after and including the 5th term all results are divisible by 3 exactly 2 times which is 9. 
20180918, 23:07  #5  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×13×73 Posts 
Quote:
0! Equals 1 not 0. ETA For k 1 to n how did the 1st term becomes 0? Last fiddled with by a1call on 20180918 at 23:12 

20180918, 23:56  #6 
Feb 2017
Nowhere
2^{2}×5^{3}×7 Posts 
The sums starting with 0! are even starting with k = 1, and greater than 2 for k > 1.
The sums starting with 1! are (as already observed) divisible by 3^2 for k > 4, and also by 11 for all k > 9. The sum 0! + ... + 29! is 2*prime, and 1! + ... + 30! is 3^2 * 11 * prime. Last fiddled with by Dr Sardonicus on 20180918 at 23:56 
20180919, 04:55  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·13·73 Posts 
The mechanics of it is:
valuation (factorial sum, prime) locks in value as soon as the valuation of the addends exceed the valuation of the running sum. So iff the running sum ever factors to a valuation higher than one (such as is the case with 3), just before the addend's valuation exceeds the valuation of the running sum, the valuation can lock in a value greater than one. General rules: https://www.mersenneforum.org/showthread.php?t=22434 Would be interesting to see what other prime factors lock in valuations of greater than one(if at all possible). Last fiddled with by a1call on 20180919 at 05:27 
20180919, 08:14  #8 
May 2018
43 Posts 
Letting and the two sequences, it is clear that if divides or then divides all the following terms of the sequences. As far as I've looked, this happens only for that divides , for that divides , for that divides and for that divides . It would be interesting to know whether this happen again, but it seems quite unlike.
I do not see any other easy properties of these number, I will think about it. Last fiddled with by ricky on 20180919 at 08:16 
20180919, 12:06  #9 
Aug 2006
1011100100101_{2} Posts 

20180919, 12:28  #10  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·13·73 Posts 
This is what I see in data section:
Quote:
With the 1st term being 0 not 1. Last fiddled with by a1call on 20180919 at 12:30 

20180919, 13:04  #11  
Jun 2003
4,703 Posts 
Quote:


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