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Old 2018-09-18, 16:06   #1
ricky
 
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Default Numbers of the form 1!+2!+3!+...

For no special reasons I started getting interested in factoring numbers of the form <br />
\sum_{n=0}^k n!<br />
 or <br />
\sum_{n=1}^k n!<br />
 . 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 2018-09-18 at 16:07
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Old 2018-09-18, 16:35   #2
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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
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Old 2018-09-18, 16:58   #3
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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.
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Old 2018-09-18, 22:53   #4
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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.
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Old 2018-09-18, 23:07   #5
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Quote:
Originally Posted by CRGreathouse View Post
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
The sequence A0074789 seems to me to be wrong.
0! Equals 1 not 0.

ETA For k 1 to n how did the 1st term becomes 0?

Last fiddled with by a1call on 2018-09-18 at 23:12
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Old 2018-09-18, 23:56   #6
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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 2018-09-18 at 23:56
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Old 2018-09-19, 04:55   #7
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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 2018-09-19 at 05:27
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Old 2018-09-19, 08:14   #8
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Letting a_k and b_k the two sequences, it is clear that if d \leq k+1 divides a_k or b_k then d divides all the following terms of the sequences. As far as I've looked, this happens only for d=2 that divides a_1 = 2, for d=3 that divides b_2 = 3, for d=9 that divides b_8 = 46233 and for d=11 that divides b_{10} = 4037913. 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 2018-09-19 at 08:16
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Old 2018-09-19, 12:06   #9
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Quote:
Originally Posted by a1call View Post
The sequence A0074789 seems to me to be wrong.
0! Equals 1 not 0.

ETA For k 1 to n how did the 1st term becomes 0?
The first term is 1! = 1. The sequence starting with 0! = 1 is A003422.
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Old 2018-09-19, 12:28   #10
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Quote:
Originally Posted by CRGreathouse View Post
The first term is 1! = 1.
This is what I see in data section:
Quote:
0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, 522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, 378011820620313, 6780385526348313, 128425485935180313, 2561327494111820313, 53652269665821260313
https://oeis.org/A007489

With the 1st term being 0 not 1.

Last fiddled with by a1call on 2018-09-19 at 12:30
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Old 2018-09-19, 13:04   #11
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Quote:
Originally Posted by a1call View Post
This is what I see in data section:


https://oeis.org/A007489

With the 1st term being 0 not 1.
If a(n) = Sum_{k=1..n} k!, what is a(0)?
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