A004231 Ackermann's sequence: n^^n := n^n^n^...^n (with n n's).

1, 1, 4, 7625597484987

Offset: 0

Author

Daniel Wild (wild(AT)edumath.u-strasbg.fr)

Keywords

• nonn

Comments

Using Knuth's arrow notation, this is n^^^2 (n-penta-2) or n^^n (n-tetra-n). - Andrew Robbins, Apr 16 2009

Comment from Trevor Green: The fourth term in this sequence has about as many digits - 8.07 * 10^153 - as the *square* of the number of protons in the universe.

We could prepend a(0) = 1 (since 0^^0 = 1, that is, the "empty power tower" gives the "empty product"). - Daniel Forgues, May 17 2013

The last 60 decimal digits of a(4) are ...67586985427238232605843019607448189676936860456095261392896. - Daniel Forgues, Jun 25 2016

From Daniel Forgues, Jul 06 2016: (Start)

a(4) has (the following number having 154 decimal digits)

80723047260282253793826303970853990300713679217387 \

43031867082828418414481568309149198911814701229483 \

451981557574771156496457238535299087481244990261351117 decimal digits.

a(4) = 4^4^4^4 = 4^

13407807929942597099574024998205846127479365820592 \

39337772356144372176403007354697680187429816690342 \

7690031858186486050853753882811946569946433649006084096,

the exponent of 4 having 155 decimal digits. (End)

The fractional part of 4^4^4*log[10](4) starts .373100157363599870..., so the first few digits of a(4) are 23610226714597313.... - Robert Israel, Jul 06 2016

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 60.

Links

• W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133, DOI:10.1007/BF01459088.
• Eric Weisstein's World of Mathematics, Ackermann Number
• Wikipedia, Knuth's up-arrow notation

Crossrefs

Cf. A027747, A008868 (which mentions an older estimate for the same number), A266200.

Main diagonal of A321312.

Programs

Maple

b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
a:= n-> b(n, n):
seq(a(n), n=0..3);  # Alois P. Heinz, Aug 22 2017

Mathematica

a[n_] := If[n == 0, 1, Nest[n^#&, n, n-1]];
Table[a[n], {n, 0, 3}] (* Jean-François Alcover, Mar 19 2019 *)