A189896 Weak Ackermann numbers: H_n(n,n) where H_n is the n-th hyperoperator.

1, 2, 4, 27

Offset: 0

Author

Max Sills, Apr 30 2011

Keywords

• nonn
• bref

Comments

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

Examples

			a(0) = succ(0) = 0 + 1 = 1, because the zeroth hyperoperation is successor.
a(1) = 1 + 1 = 2, because the first hyperoperation is addition.
a(2) = 2 * 2 = 4, because the second hyperoperation is multiplication.
a(3) = 3^3 = 27, because the third hyperoperation is exponentiation.
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.
		

Links

• M. H. Löb and S. S. Wainer, Hierarchies of number-theoretic functions. I, Archive for Mathematical Logic 13:1-2 (1970), pp. 39-51.
• Wikipedia, Hyperoperation.
• Wikipedia, Ackermann function

Crossrefs

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

Cf. A271553 ( H_n-1(n,n) ). - Natan Arie Consigli, Apr 10 2016

Formula

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

Extensions

"Weak" added to definition by Natan Arie Consigli, Apr 18 2015