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

1,1

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

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.

Example

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

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[ # ]]]&]

Prog

(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
(Magma) [ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ];  // Bruno Berselli, Jul 08 2011
(Haskell)
a046704 n = a046704_list !! (n-1)
a046704_list = filter ((== 1) . a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

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.

Sequence in context: A087521 A078403 A129945 * A367793 A089392 A089695

Adjacent sequences: A046701 A046702 A046703 * A046705 A046706 A046707

Keyword

base,nonn

Author

Felice Russo

Status

approved