While I was investigating some specific types of prime numbers I have faced with the following infinite sequence :
$1,2,8,9,15,20,26,38,45,65,112,244,303,393,560,….$
I tried to find recursive formula using Maple and it's listtorec command, so up to $393$ I got the next output:
$ f(n+3) = ((-10604990407411886564453040+8614360900967683126093782*n$ $-1437788330056801496567841*n^2-20019334790519891406942*n^3$ $+10676199651161684501481*n^4)*f(n+1)$ $+(-1637719982644311036922320-2457276199701830407970234*n$ $-480059310080505210547097*n^2+383671472063948372228234*n^3$ $-33849767081583104776903*n^4)*f(n+2))$ $/(-936042047504931985146406*n -3812415630664251269364960$ $+337414858035611215686569*n^2+50641450188283496191324*n^3$ $-8211420729473965803551*n^4) $
but when I added $560$ to list Maple sent me message FAIL.
So, my question is : how can I find pattern for this sequence if it exists ?
Best Answer
You can look up integer sequences at OEIS: http://oeis.org/A056805
So your sequence is "Numbers $n$ such that $6*10^n+1$ is prime". I assume you're looking for a formula, but if there was a closed-form expression for these numbers, we could find arbitrarily large prime numbers! The largest known prime has 12978189 digits and right now there is a 250,000 dollar prize to whoever finds a prime number with at least 1,000,000,000 digits (see http://www.eff.org/awards/coop). So if you find a formula for these numbers, please tell me.