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Smallest value of the n-fold nesting prime(prime(...(k)...) with a prime digital sum.
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%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