Wish I paid more attention at school...

[mackerel]

That's a bigger task for when I have more time. I'd have to convert the test results file to something more sheet friendly and apply the calc to it...

[mackerel]

Ok, that wasn't as bad as I thought to work out.

n		actual	predicted
10-100		2	4.0
100-1k		5	7.0
1k-10k		7	6.7
10k-100k	9	6.5
100k-1M		?	6.6
1M-10M		?	6.5

100k-435k	4	4.2
1.50M-1.82M	0	0.5

I sieved the range in two parts, 1.5M-10M initially, then below 1.5M separately. Spotted a mistake where I used p from sieve file for the higher range, as I had manually split that on multiple systems so actual depth was higher. With that corrected it slightly changes one result.

Also it seems Excel's number limit is somewhere just over 2^1000...

[R. Gerbicz]

A quick and dirty code for the sequence k*b^n+c in the range nmin<=n<=nmax:
(for ultra special sequences Mersenne/Fermat it is ofcourse a little off)

fun(k,b,c,nmin,nmax,pmax=1000)={mult=1.0;forprime(p=2,pmax,r=Mod(1,p);
cnt=0;for(i=1,p-1,r*=b;if(lift(k*r+c)!=0,cnt++));
mult*=cnt*p/(p-1)^2);
return(mult/log(b)*log(nmax/nmin))}

[robert44444uk]

Originally Posted by VBCurtis

I use a heuristic method, since the frequency of primes depends so strongly on the choice of k (thus, "high weight" and "low weight" k values).

I count the number of primes from n=1000 to 10000, then 10k to 100k, then 100k to 1M. I expect the number of primes from 1M to 10M to be similar to those counts.

Example, k=13: 4 primes in 1k-10k, 7 in 10k-100k, 2 in 100k-1M. So, I "expect" 4 or 5 primes from 1M-10M. I've tested to about 4.4M so far, and gotten rather lucky with 6 primes over 1M.

Example 2: k=99: 13 primes in 1k-10k, 12 primes in 10k-100k, 10 primes in 100k-1M. So, I "expect" 11 primes in 1M-10M. I've tested to about 3.1M, with 9 primes found (seems a bit lucky too).

When I was looking at very prime power series of this type (Sierpinski and Reisel power series) , I also used that power of 10 method (i.e. # of primes approximately the same for each 10^x range, x integer) to estimate the likely number of primes in a given range.

For the power series with the greatest number of probable primes (Reisel k= 1480472640274704456611717878515654164205) the following prp stats arise from using the power of 10 method :

n range  	# of primes
	
n 1e0 - 1e1-1 	2
n 1e1 - 1e2-1	13
n 1e2 - 1e3-1	43
n 1e3 - 1e4-1	54
n 1e4 - 1e5-1	52
n 1e5 - 1e6-1	51  *actually to n = 834442

https://www.mersenneforum.org/showthread.php?t=18407

I imagine that computers are a bit faster now, but there was no fast method for sieving or proving prp status back then. I wonder if things have moved on?