Part (in %) of Prime numbers in set of natural numbers.

Part (in %) of Prime numbers in set of natural numbers.

Postby Mblacob » Tue Jun 26, 2018 7:29 am

If we know the index [n] of given Prime Number P[n] in the set of Prime numbers, then P[n] + 1 is the number of all positive integers in the set with n Prime numbers.
It implies that ratio [tex]n/(P[n] + 1)[/tex] gives the part (in percentage) of Prime numbers in this set.
Function [tex]f(n, P) = n/(P[n] + 1)[/tex]\infty, where n = 1,2,3,...\rightarrow\infty and P[n] the Prime Number, converges to 0.05……
Example:
n = 4, P[4] = 7 \Rightarrow0.5
n = 46, P[46] = 199 \Rightarrow 0.23
n = 1053, P[1053] = 8423 \Rightarrow 0.125
n = 21944, P[21944] = 248779 \Rightarrow 0.088206447
n = 41191865, P[41191865] = 800934961 \Rightarrow 0.051429725
n = 50409172, P[50409172] = 990919439 \Rightarrow 0.05087111
n = 50848100, P[50848100] = 1000011601 \Rightarrow 0.05084751
Looks like around 5% of positive integers are Prime Numbers!?

Mblacob
 
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