A248907 -id:A248907 - cp's OEIS frontend

A032810 Numbers using only digits 2 and 3.

2, 3, 22, 23, 32, 33, 222, 223, 232, 233, 322, 323, 332, 333, 2222, 2223, 2232, 2233, 2322, 2323, 2332, 2333, 3222, 3223, 3232, 3233, 3322, 3323, 3332, 3333, 22222, 22223, 22232, 22233, 22322, 22323, 22332, 22333, 23222, 23223

Offset: 1

Clark Kimberling

Identical to A007931 with substitution of digits 2 -> 3, 1 -> 2, i.e., application of the function A048379 or A256079(n) = n + A002275(A055642(n)). - M. F. Hasler, Mar 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A020458, A143967, A248907 (permutation).

Cf. A032804-A032816 (in other bases), A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

a032810 = f 0 . (+ 1) where
   f y 1 = a004086 y
   f y x = f (10 * y + m + 2) x' where (x', m) = divMod x 2
-- Reinhard Zumkeller, Mar 18 2015

[n: n in [1..24000] | Set(Intseq(n)) subset {2, 3}]; // Vincenzo Librandi, May 27 2012

[n eq 1 select 2 else IsOdd(n) select 10*Self(Floor(n/2))+2 else Self(n-1)+1: n in [1..40]]; // Bruno Berselli, May 27 2012

Flatten[Table[FromDigits[#,10]&/@Tuples[{2,3},n],{n,5}]] (* Vincenzo Librandi, May 27 2012 *)

A032810(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*2 \\ M. F. Hasler, Mar 26 2015

def A032810(n): return int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1<<1)//9 # Chai Wah Wu, Jul 15 2023

a(n) = f(n+1, 0) with f(n, x) = if n=1 then A004086(x) else f(floor(n/2), 10*x + 2 + n mod 2). - Reinhard Zumkeller, Sep 06 2008
a(n) is Theta(n^(log_2 10)); there are about n^(log_10 2) members of this sequence up to n. - Charles R Greathouse IV, Mar 18 2010
a(n) = A007931(n) + A002275(A000523(n+1)). A055642(a(n)) = A000523(n+1). - M. F. Hasler, Mar 21 2015

A299229 {2,3}-power towers in increasing order, concatenated; see Comments.

Original entry on oeis.org

2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2

Offset: 1

Clark Kimberling, Feb 06 2018

Suppose that S is a set of real numbers. An S-power-tower, t, is a number t = x(1)^x(2)^...^x(k), where k >= 1 and x(i) is in S for i = 1..k. We represent t by (x(1), x(2), ..., x(k)), which for k > 1 is defined as (x(1), (x(2), ..., x(k))); (2,3,2) means 2^9. The number k is the *height* of t. If every element of S exceeds 1 and all the power towers are ranked in increasing order, the position of each in the resulting sequence is its *rank*.
In the following guide, "tower" means "power-tower", and t(n) denotes the n-th {2,3}-tower, represented as (x(1), x(2), ..., x(k)).
A299229: sequence of all {2,3}-towers, ranked, concatenated
A299230: a(n) = height of t(n)
A299231: all n such that t(n) has x(1) = 2
A299232: all n such that t(n) has x(1) = 3
A299233: all n such that t(n) has x(k) = 2
A299234: all n such that t(n) has x(k) = 3
A299235: a(n) = number of 2's in t(n)
A299236: a(n) = number of 3's in t(n)
A299237: a(n) = m satisfying t(m) = reversal of t(n)
A299238; a(n) = m satisfying t(m) = 5 - t(n)
A299239: all n such that t(n) is a palindrome
A299240: ranks of all t[n] in which #2's > #3's
A299241: ranks of all t[n] in which #2's = #3's
A299242: ranks of all t[n] in which #2's < #3's
A299322: ranks of t[n] in which the 2's and 3's alternate
Rectangular arrays:
A299323: row n shows ranks of towers in which #2's = n
A299324: row n shows ranks of towers in which #3's = n
A299325: row n shows ranks of towers that start with n 2's
A299326: row n shows ranks of towers that start with n 3's
A299327: row n shows ranks of towers having maximal runlength n

			As an irregular triangle, where row n contains the digits of A248907(n):
  2;
  3;
  2, 2;
  2, 3;
  3, 2;
  2, 2, 2;
  3, 3;
  3, 2, 2;
  2, 2, 3;
  2, 3, 2;
  3, 2, 3;
  3, 3, 2;
  2, 2, 2, 2;
  3, 2, 2, 2;
  2, 3, 3;
  ...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A248907, A299230-A299242, A256231, A185969, A299322-A299327.

t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
    While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
   p = p + 1; n = m]]; f = f + 1]
Flatten[Table[t[n], {n, 1, 120}]];  (* A299229 *)

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.

Original entry on oeis.org

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

A256179 Sequence of power towers in ascending order, using all possible permutations of 2's and 3's.

Original entry on oeis.org

2, 3, 4, 8, 9, 16, 27, 81, 256, 512, 6561, 19683, 65536, 43046721, 134217728, 7625597484987, 2417851639229258349412352, 443426488243037769948249630619149892803, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 1

Bob Selcoe, Mar 18 2015

nonn

a(n) is found by treating the digits of A248907(n) as power towers, so the sequence starts 2, 3, 2^2=4, 2^3=8, 3^2=9, 2^(2^2)=16, 3^3=27, 3^(2^2)=81, 2^(2^3)=256...

			a(12) = 19683 because A248907(12) = 332, and 3^(3^2) = 19683.
a(23) = 2^3^2^3 = 11423...73952 (1976 digits), because A248907(23) = 2323.

Pontus von Brömssen, Table of n, a(n) for n = 1..22
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015.

Cf. A032810, A075877, A185969, A248907, A256229.

A256179(n)=A256229(A248907[n]) \\ where A248907 is assumed to be defined as vector. - M. F. Hasler, Mar 19 2015

A256179 = A256229 o A248907 = A256229 o A032810 o A185969, i.e., a(n) = A256229(A248907(n)) = A256229(A032810(A185969(n))).

Recurrence: a(1)=2, a(2)=3, a(3)=2^2, a(4)=2^3, a(5)=3^2, a(6)=2^(2^2), a(7)=3^3, a(8)=3^(2^2), a(9)=2^(2^3), a(10)=2^(3^2), a(11)=3^(2^3), a(12)=3^(3^2); and for n>6, a(2n)=3^a(n-1), a(2n-1)=2^a(n-1). - Vladimir Reshetnikov, Mar 19 2015

More terms from M. F. Hasler, Mar 19 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

A032810 Numbers using only digits 2 and 3.

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A299229 {2,3}-power towers in increasing order, concatenated; see Comments.

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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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A256179 Sequence of power towers in ascending order, using all possible permutations of 2's and 3's.

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

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

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