[QUOTE=science_man_88;225135]I find all the terms except the first have a digital root of 4 or 1 just like the Mersenne primes >7 if we could confirm something like a pattern (I think starting at the 4th term they may take on the opposite to the series for the Mersenne primes>31) if this was ever proven true it could let us look 4 Mersenne primes in the future(that's why I'm doubtful).[/QUOTE]

Digital roots are essentially the number mod 9. Mersenne numbers other than 3 are of the form 2^(2k+1) - 1 = 4^k * 2 - 1, which is 1 mod 3. Thus Mersenne numbers other than 3 are 1, 4, or 7 mod 9. Further, the numbers are 7 mod 9 only when the exponent is a multiple of 3, which for Mersenne primes happens only with M_3 = 7. So the product of two Mersenne primes, both greater than 7, will be 1*1, 1*4, 4*1, or 4*4 mod 9, that is, will have digital root 1, 4, or 7.

You only gave the first two possibilities, but the third happens with
(2[SUP]89[/SUP] - 1)(2[SUP]107[/SUP] - 1) = 100433627766186892221372630609062766858404681029709092356097
and presumably occurs infinitely often.

[QUOTE=3.14159;225136]The law of small numbers, at work again. Unless you can definitively prove this, it is nothing more than a guess.[/QUOTE]

It seems to show that science_man_88's intuition is pretty sharp [i]vis-a-vis[/i] the strong law of small numbers, actually. :smile: Although he did miss one case, I guess because it wasn't listed in [url=http://oeis.org/classic/A165223]A165223[/url].

[QUOTE=CRGreathouse]It seems to show that science_man_88's intuition is pretty sharp [I]vis-a-vis[/I]the strong law of small numbers, actually. :smile: Although he did miss one case, I guess because it wasn't listed in A165223.
[/QUOTE]

Remember: Just because it works for three examples, doesn't mean it works forever.

[QUOTE=3.14159;225139]Remember: Just because it works for three examples, doesn't mean it works forever.[/QUOTE]

funny I checked every one it the sequence.

Also: Are there any sieving programs for n * k! + 1 ?

Nevermind: PARI does the work here. I'll set up a program which can act as an amateur sieve:

kfacsieve(x, n, a, m) = {
for(p=x,n,
if((x*n+1)%p==0, ...

[QUOTE=3.14159;225141]Also: Are there any sieving programs for n * k! + 1 ?

Nevermind: PARI does the work here. I'll set up a program which can act as an amateur sieve:

kfacsieve(x, n, a, m) = {
for(p=x,n,
if((x*n+1)%p==0, ...[/QUOTE]

I got [CODE]for(n=1,100,for(k=1,100,if(isprime(n*k!+1),print(n","k))))[/CODE]

[B]Without[/B] using (isprime(x))

Also: @CRG: I tried this:

[code] d(a,n,x,m)=for(n=a,x,if(t(n*m!+1)!=n,print("trivial composite","factor is",t(n*m!+1)));if(t(n*m!+1)==n,print(n*m!+1)))[/code]

Where t(n) is trial division up to nextprime(10^6).

I get the error: "Too many parameters in user-defined function".

I don't see where the excess params are at: I introduced no new variables into the commands. Or is it treating t as a variable?

[QUOTE=3.14159;225141]Also: Are there any sieving programs for n * k! + 1 ?[/QUOTE]

What's fixed?

[QUOTE=CRGreathouse]What's fixed?
[/QUOTE]

k is fixed, n is variable.

Reason: Primeform too slow!

[QUOTE=3.14159;225149]k is fixed, n is variable.

Reason: Primeform too slow![/QUOTE]

I got [CODE] not a function: `t'.[/CODE]

posting it into pari.

Because [B][COLOR="Red"]you have not defined t yet. I have.[/COLOR][/B]