Puzzle 1267 How many emirps are there?

				
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From 8 to 15, May, 
				2026, contributions came from Emmanuel Vantieghem, Carlos Rivera

Emmanuel wrote:

				The number of emirps with  n  digits can be found in  https://oeis.org/A152014 :

				{0, 8, 28, 204, 1406, 9538, 70474, 535578, 4192024, 33619380, 
				274890230, 2294771254}

				(for  n = 1  to  12).

				From this, we find the number of emirps < 10^n :

				{0, 8, 36, 240, 1646, 11184, 81658, 617236, 4809260, 38428640, 
				313318870, 2608090124}.

				The number of primes < 10^n  is:

				{4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 
				455052511, 4118054813, 37607912018}.

				The quotient of these numbers is (we took  n  > 1) :

				{0.32, 0.16667, 0.16599, 0.14658, 0.12151, 0.10604, 0.092959, 
				0.082443, 0.073880, 0.066752, 0.061018}

				Here is a plot 

It is clear that we have too little data to formulate a 
				decent conjecture.
But if I was forced to make a guess, I 
				would say that the curve could tend to a limit.  And I would 
				choose a limit value which depends on 10, our number base.
A 
				"good" one could be  1 / (10 ln 10).~ 0.043429448.
That would 
				lead me to the conjecture :
     Em[x] ~ x / (10 (ln 10) ( ln 
				x)).
This is widely far from  n / (log(n)^2.
But I would 
				not bet on it ! 

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				Carlos wrote:

Here are my counts asked in Q1 & my 
				comparison to the formula proposed in Q2

Note: By 
				definition an emirp exclude the palprimes:

"An emirp 
				(pronounced /ˈiːmərp/ or /ˈɛmərp/, an anadrome of prime) is a 
				prime number that results in a different prime when its decimal 
				digits (digits in base 10) are reversed.[1] This definition 
				excludes the related palindromic primes...." From 
				https://en.wikipedia.org/wiki/Emirp

I'm not sure if I'm 
				making a correct interpretation of the Simon's phrase in Q2. 
				"Compare it to n/(ln(n)*ln(n))". But here I go with my turn

					n
					
					Emirps
					Accum. Of 
					Emirps
					Palprimes
					x=10^(n+1)
					
					F=x/(ln(x)*ln(x))
					
					Acc. of emirps/F
				
					0
					0
					0
					4
					10
					
					1.89 
					
					                             -  
					
					1
					8
					8
					1
					100
					
					4.72 
					
					1.70 
				
					2
					28
					36
					15
					1000
					
					20.96 
					
					1.72 
				
					3
					204
					240
					0
					10000
					
					                   117.88
					
					2.04 
				
					4
					1406
					1646
					93
					100000
					
					                   754.45
					
					2.18 
				
					5
					9538
					11184
					0
					1000000
					
					                5,239.21
					
					2.13 
				
					6
					70474
					81658
					668
					10000000
					
					             38,492.18
					
					2.12 
				
					7
					535578
					617236
					0
					100000000
					
					           294,705.78
					
					2.09 
				
					8
					4192024
					4809260
					5172
					1000000000
					
					2,328,539.47 
					
					2.07 
				
					9
					33619380
					38428640
					0
					1E+10
					
					18,861,169.70 
					
					2.04 
				
					 n=x means the range 
					[10^x ,  10^(x+1)]
				
					Computed by my  
					"Cuenta_de_emirps.py"
				
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