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

%I A004231 #64 Mar 08 2025 09:35:16
%S A004231 1,1,4,7625597484987
%N A004231 Ackermann's sequence: n^^n := n^n^n^...^n (with n n's).
%C A004231 Using Knuth's arrow notation, this is n^^^2 (n-penta-2) or n^^n (n-tetra-n). - Andrew Robbins, Apr 16 2009
%C A004231 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.
%C A004231 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
%C A004231 The last 60 decimal digits of a(4) are ...67586985427238232605843019607448189676936860456095261392896. - _Daniel Forgues_, Jun 25 2016
%C A004231 From _Daniel Forgues_, Jul 06 2016: (Start)
%C A004231 a(4) has (the following number having 154 decimal digits)
%C A004231   80723047260282253793826303970853990300713679217387 \
%C A004231   43031867082828418414481568309149198911814701229483 \
%C A004231   451981557574771156496457238535299087481244990261351117 decimal digits.
%C A004231 a(4) = 4^4^4^4 = 4^
%C A004231   13407807929942597099574024998205846127479365820592 \
%C A004231   39337772356144372176403007354697680187429816690342 \
%C A004231   7690031858186486050853753882811946569946433649006084096,
%C A004231 the exponent of 4 having 155 decimal digits. (End)
%C A004231 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
%D A004231 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 60.
%H A004231 W. Ackermann, <a href="http://eretrandre.org/rb/files/Ackermann1928_126.pdf">Zum Hilbertschen Aufbau der reellen Zahlen</a>, Math. Ann. 99 (1928), 118-133, DOI:10.1007/BF01459088.
%H A004231 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AckermannNumber.html">Ackermann Number</a>
%H A004231 Wikipedia, <a href="http://en.wikipedia.org/wiki/Knuth&#39;s_up-arrow_notation">Knuth's up-arrow notation</a>
%p A004231 b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
%p A004231 a:= n-> b(n, n):
%p A004231 seq(a(n), n=0..3);  # _Alois P. Heinz_, Aug 22 2017
%t A004231 a[n_] := If[n == 0, 1, Nest[n^#&, n, n-1]];
%t A004231 Table[a[n], {n, 0, 3}] (* _Jean-François Alcover_, Mar 19 2019 *)
%Y A004231 Cf. A027747, A008868 (which mentions an older estimate for the same number), A266200.
%Y A004231 Main diagonal of A321312.
%K A004231 nonn
%O A004231 0,3
%A A004231 Daniel Wild (wild(AT)edumath.u-strasbg.fr)