%I #46 Sep 08 2022 08:44:56

%S 2,3,5,7,11,23,29,41,43,47,61,67,83,89,101,113,131,137,139,151,157,

%T 173,179,191,193,197,199,223,227,229,241,263,269,281,283,311,313,317,

%U 331,337,353,359,373,379,397,401,409,421,443,449,461,463,467,487,557,571,577,593

%N Additive primes: sum of digits is a prime.

%C Sum_{a(n) < x} 1/a(n) is asymptotic to (3/2)*log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - _Jonathan Sondow_, Jun 07 2012

%C Harman 2012 also shows, under a conjecture about primes in short intervals, that there are 3/2 * x/(log x log log x) terms up to x. - _Charles R Greathouse IV_, Nov 17 2014

%H Reinhard Zumkeller, <a href= "/A046704/b046704.txt" >Table of n, a(n) for n = 1..10000</a>

%H Glyn Harman, <a href= "http:// cs.uwaterloo.ca/journals/JIS/VOL15/Harman/harman2.html" >Counting primes whose sum of digits is prime</a>, J. Integer Seq., 15 (2012), Article 12.2.2.

%H Glyn Harman, <a href= "http://dx.doi.org/10.1112/blms/bds034" >Primes whose sum of digits is prime and metric number theory</a>, Bull. Lond. Math. Soc. 44:5 (2012), pp. 1042-1049.

%e The digit sums of 11 and 13 are 1+1=2 and 1+3=4. Since 2 is prime and 4 is not, 11 is a member and 13 is not. - _Jonathan Sondow_, Jun 07 2012

%p select(n -> isprime(n) and isprime(convert(convert(n,base,10),`+`)), [2,seq(2*i+1,i=1..1000)]); # _Robert Israel_, Nov 17 2014

%t Select[Prime[Range[100000]], PrimeQ[Apply[Plus, IntegerDigits[ # ]]]&]

%o (PARI) isA046704(n)={local(s,m);s=0;m=n;while(m>0,s=s+m%10;m=floor(m/10));isprime(n) & isprime(s)} \\ _Michael B. Porter_, Oct 18 2009

%o (PARI) is(n)=isprime(n) && isprime(sumdigits(n)) \\ _Charles R Greathouse IV_, Dec 26 2013

%o (Magma) [ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ]; // _Bruno Berselli_, Jul 08 2011

%o (Haskell)

%o a046704 n = a046704_list!! (n-1)

%o a046704_list = filter ((== 1). a010051. a007953) a000040_list

%o -- _Reinhard Zumkeller_, Nov 13 2011

%Y Indices of additive primes are in A075177.

%Y Cf. A046703, A119450 = Primes with odd digit sum, A081092 = Primes with prime binary digit sum, A104213 = Primes with nonprime digit sum.

%Y Cf. A007953, A010051; intersection of A028834 and A000040.

%K base,nonn

%O 1,1

%A _Felice Russo_