[seqfan] Re: Least Prime Factor Counting Function

[Matthijs Coster]

Dear William,

In approximation your answer is correct, however the numerators are larger.

2: fraction 1/2 >>  Sum 1/2 = 1 - 1/2
3: fraction 1/(2*3) >> Sum 2/3 =  1-2/6
5: fraction 1/(3*5) >> Sum 11/15 = 1 - 8/30
7: fraction 4/(3*5*7) >> Sum 27/35 = 1 - 48/210
etc.


In the last column has been written 1 - phi(Prod(primes))/Prod(primes)


Hartelijke groeten
Matthijs Coster

Op 11-2-2012 12:42, William Keith schreef:
> 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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