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• Hi Everybody Finally fixed the computer,so I would like to belatedly thank Jack Brennen, Phil Carmody, Joe McClean and Chris Nash for their comments on my
Aug 29, 2001 1 of 3
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Hi Everybody
Finally fixed the computer,so I would like to belatedly thank Jack
Brennen, Phil Carmody, Joe McClean and Chris Nash for their comments on
my Sierpinski and Little Fermat posts. Gentlemen, I do appreciate your
taking the time to answer questions I should have been able to answer for
myself by a more diligent search of the web. I do try but sometimes find
it frustratingly difficult.
For example Ive noticed that for certain primes P, the number of
composite ODD integers up to and including P is evenly divisible by the
number of primes up to and including P. The same for EVEN integers. Put
in different words, the number of ODDS or EVENS equals an integer N times
the number of primes. An approximation to N is the integer part of (LN(P)
 2)/2 for odds and the integer part of (LN(P)  1)/2 for evens.
Clearly this pertains to the Nth Prime and the Prime Counting Function
categories and surely someone investigating these fields must have also
noticed this, but Ive had no luck finding any references. Maybe its so
trivial its not worth mentioning. Here are some specifics for anyone
interested:
A represents the number of consecutive primes that exist up to and
including P, labeling the prime 2 as number 1.
B represents the number of ODD COMPOSITE integers up to and including P,
labeling the odd integer 1 as number 1. B = ((P+1) / 2) - A
C represents the number of EVEN integers up to P, labeling the even
integer 2 as number 1. C = (P-1) / 2
RO represents the ratio B / A
RE represents the ratio C / A
From a list of 100,000 consecutive primes which I downloaded and assume
to be accurate, I sifted out the following data according to the criteria
that RO and RE be integers:
For primes having integer ROs:
Primes 3, 5, 1091, 8423, 64579, 64609, 64709, 481043, 481067
As 2, 3, 182, 1053, 6458, 6461, 6471, 40087, 40089
Bs 0, 0, 364, 3159, 25832, 25844, 25884, 200435, 200445
ROs 0, 0, 2, 3, 4, 4, 4, 5, 5
For primes having integer REs:
Primes 11, 13, 1087, 64591, 64601, 64661
As 5, 6, 181, 6459, 6460, 6466
Cs 5, 6, 543, 32295, 32300, 32330
REs 1, 1, 3, 5, 5, 5
Is it possible to prove that the supply of such Ps is infinite? Is it
possible to find a P such that an integer RO = an integer RE? Are all
integer REs odd? And a lot of other questions. Im hoping someone having
access to a larger database will extend my range. I ended with the prime
1,299,827 which is prime number 100,008. Id appreciate any comments and
thanks everyone.
Bill Sindelar
• Hi Everybody Mea Maxima Culpa. Please make the following correction to my post Prime Counting Function . In the last paragraph instead of the question Is it
Aug 30, 2001 1 of 3
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Hi Everybody
Mea Maxima Culpa. Please make the following correction to my post "Prime
Counting Function". In the last paragraph instead of the question "Is it
possible to find a P such that an integer RO = an integer RE?" I should
have asked "Is it possible to find a P such that A evenly divides both B
and C?" I think it makes it a bit more understandable as to what I mean.
My abject apology. Thanks again everyone.
Bill Sindelar
• According to Prime Number Theorom a) pi_a(n) ~ Li(n) above is the improvement over b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336 For those who cannot compute the
Sep 7, 2005 1 of 3
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According to Prime Number Theorom

a) pi_a(n) ~ Li(n)

above is the improvement over

b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336

For those who cannot compute the logarithmic integral Li(n) the
following counting functions yields better results than b)

c) pi_c(n) ~ n{1 +0.5x +sqrt[x^2/4 -x-1 -1/(4x^2)]}/(2 x)
where x=ln(n)

let n=10^k below the differences to the actual pi(10^k)

k ; a) ; b) ; c)

3 ; 9 ; 3 ; 9
6 ; 129 ; 43 ; 105
9 ; 1700 ; 69207 ; 312
12 ; 3.8 10^4 ; 6.1 10^7 ; -1.1 10^5
15 ; 1.1 10^6 ; 4.6 10^10 ; -2.5 10^7
18 ; 2.2 10^7 ; 3.5 10^13 ; -2.5 10^8
21 ; 5.9 10^8 ; 2.7 10^16 ; -4.1 10^12
23 ; 7.2 10^9 ; 2.3 10^18 ; -4.3 10^14

pi_a(n) ~ Li(n) is still the best

Has any one seen the approximation pi_c() before

I discovered it by investigating the number of primes
between a^b and a^(b+1)

regards
Anton
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