|
%I #23 May 13 2013 01:54:21
%S 2,3,5,7,11,23,29,41,43,47,61,83,101,113,131,137,151,173,191,223,227,
%T 241,263,281,311,313,317,331,353,401,421,443,461,599,601,641,797,821,
%U 887,911,977,1013,1019,1031,1033,1051,1091,1103,1109,1123,1163,1181,1213,1217
%N Primes p whose digit sum s(p) and iterated digit sum s(s(p)) are also prime.
%C Sum_{a(n) < x} 1/a(n) is asymptotic to (9/4)*log(log(log(log(x)))) as x -> infinity; see Harman (2012). Thus the sequence is infinite.
%C The first member not in A070027 is 59899999.
%C A046704 is primes p with s(p) also prime. A070027 is primes p with all s(p), s(s(p)), s(s(s(p))), ... also prime. A104213 is primes p with s(p) not prime. A207293 is primes p with s(p) also prime, but not s(s(p)). A213354 is primes p with s(p) and s(s(p)) also prime, but not s(s(s(p))). A213355 is smallest prime p whose k-fold digit sum s(s(..s(p)).)..)) is also prime for all k < n, but not for k = n.
%H Charles R Greathouse IV, <a href="/A207294/b207294.txt">Table of n, a(n) for n = 1..10000</a>
%H G. Harman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Harman/harman2.pdf">Counting Primes whose Sum of Digits is Prime</a>, J. Integer Seq., 15 (2012), Article 12.2.2.
%e 59899999 and s(59899999) = 5+9+8+9+9+9+9+9 = 67 and s(s(59899999)) = s(67) = 6+7 = 13 are all primes, so 59899999 is a member. But s(s(s(59899999))) = s(13) = 1+3 = 4 is not prime, so 59899999 is not a member of A070027.
%t Select[Prime[Range[200]], PrimeQ[Apply[Plus, IntegerDigits[#]]] && PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]] &]
%o (PARI) select(p->my(s=sumdigits(p));isprime(s)&&isprime(sumdigits(s)), primes(1000)) \\ _Charles R Greathouse IV_, Jun 10 2012
%Y Cf. A046704, A070027, A104213, A207293, A213354, A213355.
%K nonn,base
%O 1,1
%A _Jonathan Sondow_, Jun 09 2012
|
