A046704 Additive primes: sum of digits is a prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Author

Felice Russo

Keywords

• base
• nonn

Comments

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

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

Examples

			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
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Glyn Harman, Counting primes whose sum of digits is prime, J. Integer Seq., 15 (2012), Article 12.2.2.
• Glyn Harman, Primes whose sum of digits is prime and metric number theory, Bull. Lond. Math. Soc. 44:5 (2012), pp. 1042-1049.

Crossrefs

Indices of additive primes are in A075177.

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

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

Programs

Haskell

a046704 n = a046704_list !! (n-1)
a046704_list = filter ((== 1) . a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

Magma

[ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ];  // Bruno Berselli, Jul 08 2011

Maple

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

Mathematica

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

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

PARI

is(n)=isprime(n) && isprime(sumdigits(n)) \\ Charles R Greathouse IV, Dec 26 2013