A248907 Numbers consisting only of digits 2 and 3, ordered according to the value obtained when the digits are interspersed with (right-associative) ^ operators.

2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332, 2222, 3222, 233, 333, 2322, 3322, 2223, 3223, 2232, 3232, 2323, 3323, 2332, 3332, 22222, 32222, 23222, 33222, 2233, 3233, 2333, 3333, 22322, 32322, 23322, 33322, 22223, 32223, 23223, 33223, 22232, 32232

Offset: 1

Author

Vladimir Reshetnikov and Reinhard Zumkeller, Mar 18 2015

Keywords

• nonn
• easy

Comments

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

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

Links

• Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015.
• Vladimir Reshetnikov et al., Power towers of 2 and 3 - looking for a proof, on StackExchange.com, Mar 19 2015

Crossrefs

Cf. A032810, A185969, A256179, A256229.

For another version, see A299229 (each digit is a separate term).

Programs

Haskell

a248907 = a032810 . a185969

Mathematica

ClearAll[a, p];
p[d_, n_] := d 10^IntegerLength[n] + n;
a[n_ /; n <= 12] := a[n] = {2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332}[[n]];
a[n_ /; OddQ[n]]  := a[n] = p[2, a[(n - 1)/2]];
a[n_] := a[n] = p[3, a[(n - 2)/2]];
Array[a, 100]

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

Formula

a(n) = A032810(A185969(n)).

Extensions

Edited by M. F. Hasler, Mar 21 2015