A256179 -id:A256179 - cp's OEIS frontend

A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 12, 13, 15, 23, 10, 14, 19, 27, 16, 24, 17, 25, 20, 28, 21, 29, 31, 47, 39, 55, 18, 26, 22, 30, 35, 51, 43, 59, 32, 48, 40, 56, 33, 49, 41, 57, 36, 52, 44, 60, 37, 53, 45, 61, 63, 95, 79, 111, 71, 103, 87, 119, 34, 50, 42, 58, 38

Offset: 1

Vladimir Reshetnikov, Feb 07 2011

			a(6) =  7; tower(7)  = 2^2^2 = 2^4 =  16.
a(7) =  6; tower(6)  = 3^3   =        27.
a(8) = 11; tower(11) = 3^2^2 = 3^4 =  81.
a(9) =  8; tower(8)  = 2^2^3 = 2^8 = 256.

Alois P. Heinz, Table of n, a(n) for n = 1..14335
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015
Index entries for sequences that are permutations of the natural numbers

Cf. A032810, A081241, A248907, A256179, A256231, A375374 (colexicographic instead of lexicographic order).

a(2*n-1) = A081241(2*A081241(a(n-1))+1) and a(2*n) = A081241(A081241(a(2*n-1))+1) for n >= 7. - Pontus von Brömssen, Aug 10 2024

More terms from Alois P. Heinz, Apr 05 2011

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

Original entry on oeis.org

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

Vladimir Reshetnikov and Reinhard Zumkeller, Mar 18 2015

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

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

Cf. A032810, A185969, A256179, A256229.

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

Haskell
```
a248907 = a032810 . a185969
```

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]

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

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

Edited by M. F. Hasler, Mar 21 2015

A256229 Powering the decimal digits of n (right-associative) with 0^0 = 1 by convention.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 1

M. F. Hasler, Mar 19 2015

See A075877 for the left-associative version (which grows much more slowly). Usually the "^" operator is considered right-associative (so this is the "natural" version), i.e., a^b^c = a^(b^c) since (a^b)^c could be written a^(b*c) instead, while there is no such simplification for a^(b^c).

If n's first digit is succeeded by an odd number of consecutive 0's, a(n) is 1. If it is by an even number, a(n) is the first digit of n (A000030). - Alex Costea, Mar 27 2019

			a(253) = 2^5^3 = 2^(5^3) = 2^125 = 42535295865117307932921825928971026432.

Alois P. Heinz, Table of n, a(n) for n = 1..237
Wikipedia, Zero to the power of zero

Cf. A007953, A007954, A075877, A133500, A185969, A248907, A256179.

a:= proc(n) local m, r; m, r:= n, 1;
      while m>0 do r:= irem(m, 10, 'm')^r od; r
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 19 2015

Power @@ IntegerDigits@ # & /@ Range@ 64 /. Indeterminate -> 1 (* Michael De Vlieger, Mar 21 2015 *)

A256229(n,p=1)={until(!n\=10,p=(n%10)^p);p}

def A256229(n):
    y = 1
    for d in reversed(str(n)):
        y = int(d)**y
    return y # Chai Wah Wu, Mar 21 2015

a(n) = A075877(n) for n < 212.
a(n) = A133500(n) for n < 100.
a(10n+1) = a(n).

Incorrect comments deleted by Alex Costea, Mar 24 2019

A075877 Powering the decimal digits of n (left-associative).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 1

Reinhard Zumkeller, Oct 16 2002

See A256229 for the (maybe more natural) "right-associative" variant, a(xyz)=x^(y^z). a(n) = A256229(n) for n < 212 (up to 210, according to the 2nd formula which also holds for A256229), but (2^1)^2 = 4 while 2^(1^2) = 1. - M. F. Hasler, Mar 22 2015

			a(253) = (2^5)^3 = 32^3 = 32768.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A007954.

Cf. A185969, A248907, A256179.

a075877 n = if n < 10 then n else a075877 (n `div` 10) ^ (n `mod` 10)
-- Reinhard Zumkeller, May 27 2013

A075877(n)=if(n>9,A075877(n\10)^(n%10),n) \\ M. F. Hasler, Mar 19 2015

a(n) = if n < 10 then n else a(floor(n\10))^(n mod 10).
a(n) = 1 iff the initial digit is 1 or n contains a 0 (i.e., A055641(n) > 0 or A000030(n) = 1);
a(A011540(n)) = 1.
a(n) = A133500(n) for n <= 99. - Reinhard Zumkeller, May 27 2013

Formula corrected by Reinhard Zumkeller, May 27 2013

Edited by M. F. Hasler, Mar 22 2015

cp's OEIS Frontend

A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

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

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A256229 Powering the decimal digits of n (right-associative) with 0^0 = 1 by convention.

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Maple

Mathematica

PARI

Python

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A075877 Powering the decimal digits of n (left-associative).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Haskell

PARI

Formula

Extensions