A162253 Smallest value of the n-fold nesting prime(prime(...(k)...)) with a prime digital sum.

2, 3, 5, 11, 1787, 5381, 5381, 5381, 648391, 648391, 414507281407, 414507281407

Offset: 1

Author

Cino Hilliard, Jun 29 2009

Keywords

• nonn
• base
• more

Comments

n-deep nestings prime(prime(...(prime(k))...)) = prime^n(k) can be arranged in a table T(n,k),

2 3 5 7 11 13 : A000040, n=0

3 5 11 17 31 41 : A006450, n=1

5 11 31 59 127 179 : A038580, n=2

11 31 127 277 709 1063 : A049090

31 127 709 1787 5381 8527 : A049203

127 709 5381 15299 52711 87803 : A049202

a(n) is the leftmost value in the n-th row (the one with the smallest k) with a digit sum which is prime.

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.

Examples

			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.
		

Programs

PARI

for(j=1,12,print(j","sodip2(100,j)","));
sodip2(n,m) = \\multiple nesting of prime(prime(prime..(n)
{
local(s=0,a,x,y,j,p);
for(x=1,n,
for(i=1,m,p=prime2(p));
a=eval(Vec(Str(p)));
y=sum(j=1,length(a),a[j]);
if(isprime(y),return(p));
)
}

Formula

{min A000040^n(k): A000040^n(k) in A028834}. - R. J. Mathar, Jul 16 2009

Extensions

Definition rephrased by R. J. Mathar, Jul 16 2009