1
$\begingroup$

I counted the number of primes, such that the digit sum is also prime with the following PARI/GP-program and compared it with the number of primes below the powers of $10$ upto $10^9$

? for(k=1,9,x=0;y=0;for(j=1,10^k,if(isprime(j)==1,y=y+1;if(isprime(sumdigits(j))
==1,x=x+1)));print(k,"  ",x,"  ",y))
1  4  4
2  14  25
3  89  168
4  590  1229
5  3883  9592
6  30123  78498
7  246982  664579
8  2163899  5761455
9  18661619  50847534  

I do not know, if there is a name for the number of primes below $x$, such that the digitsum is also prime, so I call the function $u(x)$.

  • What is known about the asymptotic behaviour of $u(x)$. In particular, does the limit $$\lim_{x\rightarrow \infty} \frac{u(x)}{\pi(x)}$$ , where $\pi(x)$ is the prime counting function, exist ?
$\endgroup$
  • 1
    $\begingroup$ We would expect that limit to be $0$, that is, the prime-digit-sum primes should have relative density $0$ inside the primes (although I doubt this has been proved). Indeed, the average digit sum of a prime near $x$ is about $4.5(\log x)/\log 10$, and the probability of a random integer that size being prime is about $1/\log\log x$ or so. So we expect that $u(x) \sim \pi(x)/\log\log x$. (We might need to multiply by some constant, reflecting local conditions: the digit sum will never be a multiple of $3$, for example.) $\endgroup$ – Greg Martin Sep 12 '15 at 16:15
  • $\begingroup$ Interesting question! As a total aside, for(k=1,9,x=0;forprime(p=1,10^k,if(isprime(sumdigits(p)),x++));print(k," ",x," "primepi(10^k))) runs much faster (Pari 2.6+). $\endgroup$ – DanaJ Sep 12 '15 at 16:34
0
$\begingroup$

If $j$ has $m$ digits in base $b$, then the size of $sumdigits(j)$ is about $mb/2$, so the probability that $sumdigits(j)$ is prime is about $1/\ln(mb/2)$.

Since there are about $\frac{b^m}{m\ln(b)}$ primes with $m$ digits, there should be about $\frac{b^m}{m\ln(b)\ln(mb/2)}$ primes with $m$ digits with sumdigits also prime.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.