%I #11 Mar 12 2015 00:20:29
%S 2,3,5,11,1787,5381,5381,5381,648391,648391,414507281407,414507281407
%N Smallest value of the n-fold nesting prime(prime(...(k)...) with a prime digital sum.
%C n-deep nestings prime(prime(....(prime(k)..) = prime^n(k) can be arranged in a table T(n,k),
%C ..2...3....5.....7....11....13 : A000040, n=0
%C ..3...5...11....17....31....41 : A006450, n=1
%C ..5..11...31....59...127...179 : A038580, n=2
%C .11..31..127...277...709..1063 : A049203
%C .31.127..709..1787..5381..8527 : A049202
%C 127.709.5381.15299.52711.87803
%C a(n) is the leftmost value in the n-th row (the one with the smallest k) with a digit sum which is prime.
%C In order to generate the entries a(11) and a(12), prime2() was used which reads a large 880 gigabyte file of all primes < 10^12.
%F {min A000040^n(k): A000040^n(k) in A028834}. - _R. J. Mathar_, Jul 16 2009
%e 1st nesting is prime(1) = 2 which has a prime digit sum: a(0). The second nesting is prime(prime(1)) = 3, which has a prime digits sum: a(1)=3. The 3rd and 4th nesting also succeed for k=1 while the fifth nesting prime(prime(prime(prime(prime(4))))) = 1787 is the first occurrence of sum of digits is prime. Here nesting for k = 1,2,3 does not sum to a prime number.
%o (PARI) for(j=1,12,print(j","sodip2(100,j)","));
%o sodip2(n,m) = \\multiple nesting of prime(prime(prime..(n)
%o {
%o local(s=0,a,x,y,j,p);
%o for(x=1,n,
%o for(i=1,m,p=prime2(p));
%o a=eval(Vec(Str(p)));
%o y=sum(j=1,length(a),a[j]);
%o if(isprime(y),return(p));
%o )
%o }
%K nonn,base,more
%O 1,1
%A _Cino Hilliard_, Jun 29 2009
%E Definition rephrased by _R. J. Mathar_, Jul 16 2009