%I #75 Nov 15 2022 17:51:53

%S 1,2,4,27

%N Weak Ackermann numbers: H_n(n,n) where H_n is the n-th hyperoperator.

%C The next term, a(4), has about 8*10^153 decimal digits. - _Charles R Greathouse IV_, Nov 15 2022

%H M. H. Löb and S. S. Wainer, <a href="http://dx.doi.org/10.1007/BF01967649">Hierarchies of number-theoretic functions. I</a>, Archive for Mathematical Logic 13:1-2 (1970), pp. 39-51.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperoperation">Hyperoperation</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann function</a>

%F a(n) = H_n(n, n), where H_n the hyperoperation indexed by n.

%e a(0) = succ(0) = 0 + 1 = 1, because the zeroth hyperoperation is successor.

%e a(1) = 1 + 1 = 2, because the first hyperoperation is addition.

%e a(2) = 2 * 2 = 4, because the second hyperoperation is multiplication.

%e a(3) = 3^3 = 27, because the third hyperoperation is exponentiation.

%e a(4) = 4^4^4^4 = 4^(4^(4^4)) = 4^(4^256), because the fourth hyperoperation is tetration. The term is too big to be included: log_2(a(4)) = 2^513.

%Y For H_n(x,x) with fixed x, cf. A054871 (x=3, shifted), A141044 (x=1), A253855 (x=4, shifted), A255176 (x=2), A256131 (x=10, shifted). - _Danny Rorabaugh_, Oct 20 2015

%Y Cf. A271553 ( H_n-1(n,n) ). - _Natan Arie Consigli_, Apr 10 2016

%K nonn,bref

%O 0,2

%A _Max Sills_, Apr 30 2011

%E "Weak" added to definition by _Natan Arie Consigli_, Apr 18 2015