Fulsorials
Hi all,
There are Factorials, primorials, multfactorials. But as far as I know the following is not coined. I would like to introduce [B]Fulsorials[/B] to you.:smile: You can calculate [B]Fulsorials[/B] by: Multiplying 2 consecutive integers, Then multiplying theproduct by thatproduct(+) 1 And continue indefinitely. Every new multiplication will be by a new coprime and No primality test is required. It could be used for finding random large factors to prime candidates without having to prove those factors primes. As an example of [B]Fulsorials[/B]: [B]6$=2*3*7*43*1807*3263443[/B] It can also be useful for finding large PRPs.:smile: 
Lovely title!

Somehow I knew you'd like it. I gave the title more thought than the subject.:smile:

[QUOTE=a1call;453496]Somehow I knew you'd like it. I gave the title more thought than the subject.:smile:[/QUOTE]
not really because there are alternatives [url]https://en.wikipedia.org/wiki/Falling_and_rising_factorials[/url] allows two types of factorials for example. edit: and there's [url]https://en.wikipedia.org/wiki/Gamma_function[/url] as an extension. etc. 
Here is a hopefully useful code for finding random factors (have not tested it myself yet, but expect a decent performance). Tweak the for and while loop parameters to suit your needs.
Also would appreciate large integers posted here for trial runs. Thank you in advance. [CODE]print("\nBMT100AAlternativeFactorials=FalsorialsRandomFactors.gp\n") allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() n=12345679001 isprime(n) for (i=3,19,{ falsorial=i; while(falsorial<10^10, falsorial=falsorial*(falsorial1); theGcd=gcd(falsorial,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); ); }) print("**** End of Run ****")[/CODE] 
[QUOTE=a1call;453494]Hi all,
There are Factorials, primorials, multfactorials. But as far as I know the following is not coined. I would like to introduce [B]Fulsorials[/B] to you.:smile: You can calculate [B]Fulsorials[/B] by: Multiplying 2 consecutive integers, Then multiplying theproduct by thatproduct(+) 1 And continue indefinitely. Every new multiplication will be by a new coprime and No primality test is required. It could be used for finding random large factors to prime candidates without having to prove those factors primes. As an example of [B]Fulsorials[/B]: [B]6$=2*3*7*43*1807*3263443[/B] It can also be useful for finding large PRPs.:smile:[/QUOTE] A specific type of "Fulsorials" are [URL="https://en.wikipedia.org/wiki/Sylvester%27s_sequence"]Sylvester's Sequence[/URL]. You have a much more general idea of this. 
[QUOTE=carpetpool;453511]A specific type of "Fulsorials" are [URL="https://en.wikipedia.org/wiki/Sylvester%27s_sequence"]Sylvester's Sequence[/URL]. You have a much more general idea of this.[/QUOTE]
Thank you for that carpetpool. I am only 137 years too late.:picard: 
[URL]https://factordb.com/index.php?id=1100000000905790309[/URL]
[CODE]print("\nBMT100CAlternativeFactorials=FalsorialsRandomFactors.gp\n") allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() n= 4883945163367692991 isprime(n) for (i=3,19^4,{ falsorial=i; while(falsorial<10^100000, falsorial=falsorial*(falsorial1);\\print(falsorial); theGcd=gcd(falsorial,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); theGcd=gcd(falsorial+1,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); ); }) print("**** End of Run ****") [/CODE] ETA: [url]https://factordb.com/index.php?id=1100000000905788578[/url] [CODE] n= 254035168468567119979994968319537 %2 = 254035168468567119979994968319537 (00:10) gp > isprime(n) %3 = 0 (00:10) gp > for (i=3,19^4,{ falsorial=i; while(falsorial<10^100000, falsorial=falsorial*(falsorial1);\\print(falsorial); theGcd=gcd(falsorial,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); theGcd=gcd(falsorial+1,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); ); }) *** Found a factor: 41 (00:10) gp > print("**** End of Run ****") **** End of Run ****[/CODE] 
for others that may be interested you have a lot of alternatives:
[url]https://en.wikipedia.org/wiki/Factorial[/url] talks of hyperfactorials and superfactorials [url]https://en.wikipedia.org/wiki/Alternating_factorial[/url] is another one and the bottom links on some of these include: [url]https://en.wikipedia.org/wiki/Bhargava_factorial[/url] and [url]https://en.wikipedia.org/wiki/Exponential_factorial[/url] 
Sounds like someone needs to do some prime hunting (and not me this time).

I am not sure who you are referring to and I can't speak for Mr Sylvester. But I wouldn't have a clue how to fully factor any of the larger terms. So if anyone feels any off this is of any use you have my blessings to use them.
From Carpetpool's link: [QUOTE]The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. [/QUOTE] And my factoring code seems to miss a lot of larger prime factors. But I still think it can be useful for large PRPs. 
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