A256229 -id:A256229 - cp's OEIS frontend

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

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

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

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

A321990 Positive numbers for which the product of digits is equal to the power tower of digits (right-associative).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 21, 22, 31, 41, 51, 61, 71, 81, 91, 111, 211, 221, 311, 411, 511, 611, 711, 811, 911, 1111, 2111, 2211, 2412, 3111, 3313, 4111, 4212, 5111, 6111, 6213, 7111, 8111, 8214, 9111, 11111, 21111, 22111, 22212, 24112, 24121, 28128, 28144

Offset: 1

Michal Gren, Nov 23 2018

Positive numbers k such that A007954(k) = A256229(k).
All numbers of the form xx1x...x with x x's are terms, as are numbers of the form xxx1x...x with x^x x's, and so on.
If the first two digits of a number are x,y, respectively, and if (x^(y-1))/y is a positive integer, then the number of the form xy1(...), where (...) is a sequence of digits whose product is (x^(y-1))/y, is a term. - Michal Gren, Nov 29 2018

			6213 is a term since 6^2^1^3 = 6*2*1*3 = 36.
8^4 = 4096. 8*4 = 32. So 841 followed by any sequence of digits whose product is 4096/32 = 128 is in the sequence. - _David A. Corneth_, Nov 28 2018

Michal Gren, Table of n, a(n) for n = 1..10000

Cf. A001055, A007954, A256229.

aQ[n_] := Module[{digits = IntegerDigits[n]}, If[MemberQ[digits, 0], False, Power@@digits == Times@@digits]]; Select[Range[1000], aQ] (* for small terms, or: *) aQ[n_] := Module[{d=IntegerDigits[n]}, If[MemberQ[d, 0], Return[False]]; p = Times@@d; If[MemberQ[d, 1], If[d[[1]]==1, Return[p==1]]; d = d[[1 ;; FirstPosition[d, 1][[1]]-1]]]; Do[p = Log[d[[i]], p], {i,1,Length[d]}]; p==1]; Select[Range[1000], aQ] (* Amiram Eldar, Nov 24 2018 *)

a007954(n) = my(d=digits(n)); vecprod(d);
f256229(n, pd)= my(p=1); until(!n\=10, p=(n%10)^p; if (p>pd, return (-p))); p;
isok(k) = my(pd = a007954(k)); pd == f256229(k, pd); \\ Michel Marcus, Nov 25 2018

cp's OEIS Frontend

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

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PARI

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

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Programs

Haskell

PARI

Formula

Extensions

A321990 Positive numbers for which the product of digits is equal to the power tower of digits (right-associative).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI