[seqfan] Re: Least Prime Factor Counting Function

[fransis fransis]

Hi, I noticed that the number of 7s in the list of smallest
prime factors of numbers is exactly 1/6 of the total of the values greater
than 7
in a periodic fashion.

For exactly this arguments of the lpf(i):
30,53,54,55,56,57,58,79,80,81,82,107,108,187,188,189,190,211,212,213,214,215,216,239
the ratio is 1/6.
And it seems so for 30+ k210,53+k210,........,239+k210
So is it possible that this is true for all primes,id est Pi(p)/Pi(>p)==
1/(p-1) for a fixed number of arguments of lpf and a period Prod(i<p)?.

2012/2/11 William Keith <william.keith at gmail.com>

> The number 7 will be the smallest prime factor of every 7th number that is
> not divisible by 2, 3, or 5.  Thus the number of 7s in the list of smallest
> prime factors of numbers in an interval from x to x+c will be periodic with
> period 210 (2*3*5*7), with a different function for each c.  Since the
> periods for each prime are subperiods of the periods for larger primes, the
> correlations over large intervals will be very strong.
>
> I'm not sure what you want concerning telescoping series.  The fraction
> will be roughly a product of 1/(2*3*5*7) times c, with fractional
> corrections based on your starting point x.
>
> Cordially,
> William Keith
>
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