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%I #15 Jun 16 2012 01:02:50
%S 13,67,59899999
%N Smallest prime p whose k-fold digit sum s(s(..s(p)..)) is also prime for all k < n, but whose n-fold digit sum is not prime.
%C The next term has more than 6655555 digits, because s(a(4)) >= a(3) = 59899999 and 59899999/9 > 6655555.
%C s(a(2)) = a(1) and s(a(3)) = a(2). Can one prove that s(a(n+1)) = a(n), for all n > 0? (Cf. formula below.) If yes, then a(n+1) is the smallest prime p with s(p) = a(n).
%C A046704 is primes p with s(p) also prime. A207294 is primes p with s(p) and s(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))).
%F a(n) <= s(a(n+1)). (Proof: a(n) and s(a(n+1)) share the same property, but a(n) is minimal.)
%e s(13) = 1+3 = 4 is not prime, and s(p) is prime if p < 13, so a(1) = 13.
%e s(67) = 6+7 = 13 is prime, but s(s(67)) = s(13) = 1+3 = 4 is not prime, and no p < 67 has this property, so a(2) = 67.
%e s(59899999) = 5+9+8+9+9+9+9+9 = 67 and s(s(59899999)) = s(67) = 6+7 = 13 are prime, but s(s(s(59899999))) = s(13) = 1+3 = 4 is not prime, and no p < 59899999 has this property, so a(3) = 59899999.
%Y a(1) = A104213(1), a(2) = A207293(1), a(3) = A213354(1).
%Y Cf. A046704, A070027, A104213, A207293, A207294, A213354.
%K nonn,base,more,bref
%O 1,1
%A _Jonathan Sondow_, Jun 10 2012