%I #49 Aug 09 2024 09:46:52
%S 2,3,22,23,32,222,33,322,223,232,323,332,2222,3222,233,333,2322,3322,
%T 2223,3223,2232,3232,2323,3323,2332,3332,22222,32222,23222,33222,2233,
%U 3233,2333,3333,22322,32322,23322,33322,22223,32223,23223,33223,22232,32232
%N Numbers consisting only of digits 2 and 3, ordered according to the value obtained when the digits are interspersed with (right-associative) ^ operators.
%C A256179(n) is found by treating the digits of a(n) as power towers. So for example, a(11) = 323, so A256179(11) = 6561 because 3^(2^3) = 6561. - _Bob Selcoe_, Mar 18 2015
%C This is a permutation of the list A032810 (numbers having only digits 2 and 3) in the sense that is a list with exactly the same terms but in different order, namely such that the ("power tower") function A256229 yields an increasing sequence. The permutation of the indices is given by A185969, cf. formula. - _M. F. Hasler_, Mar 21 2015
%H Vladimir Reshetnikov, <a href="http://list.seqfan.eu/oldermail/seqfan/2015-March/014595.html">2-3 sequence puzzle</a>, SeqFan list, Mar 18 2015.
%H Vladimir Reshetnikov et al., <a href="http://math.stackexchange.com/questions/1196327/power-towers-of-2-and-3-looking-for-a-proof">Power towers of 2 and 3 - looking for a proof</a>, on StackExchange.com, Mar 19 2015
%F a(n) = A032810(A185969(n)).
%t ClearAll[a, p];
%t p[d_, n_] := d 10^IntegerLength[n] + n;
%t a[n_ /; n <= 12] := a[n] = {2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332}[[n]];
%t a[n_ /; OddQ[n]] := a[n] = p[2, a[(n - 1)/2]];
%t a[n_] := a[n] = p[3, a[(n - 2)/2]];
%t Array[a, 100]
%o (Haskell)
%o a248907 = a032810 . a185969
%o (PARI) vecsort(A032810,(a,b)->A256229(a)>A256229(b)) \\ Assuming that A032810 is defined as a vector. Append [1..N] if the vector A032810 has too many (thus too large) elements: recall that 33333 => 3^(3^(3^(3^3))). - _M. F. Hasler_, Mar 21 2015
%Y Cf. A032810, A185969, A256179, A256229.
%Y For another version, see A299229 (each digit is a separate term).
%K nonn,easy
%O 1,1
%A _Vladimir Reshetnikov_ and _Reinhard Zumkeller_, Mar 18 2015
%E Edited by _M. F. Hasler_, Mar 21 2015