A299229 -id:A299229 - 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

A299240 Ranks of {2,3}-power towers in which #2's > #3's; see Comments.

Original entry on oeis.org

1, 3, 6, 8, 9, 10, 13, 14, 17, 19, 21, 27, 28, 29, 30, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 51, 55, 56, 57, 58, 59, 60, 61, 63, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 103, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Clark Kimberling, Feb 07 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*. See A299229 for a guide to related sequences.

This sequence together with A299241 and A299242 partition the positive integers.

			The first six terms are the ranks of these towers: t(1) = (2), t(3) = (2,2), t(6) = (2,2,2), t(8) = (3,2,2), t(9) = (2,2,3), t(10) = (2,3,2).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299241, A299242.

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]
Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &];  (* A299241 *)
Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];   (* A299242 *)

A299241 Ranks of {2,3}-power towers in which #2's = #3's; see Comments.

Original entry on oeis.org

4, 5, 18, 20, 22, 23, 25, 31, 62, 74, 76, 77, 82, 84, 85, 90, 92, 93, 96, 97, 99, 104, 105, 107, 128, 129, 131, 135, 238, 246, 250, 252, 253, 294, 298, 300, 301, 306, 308, 309, 312, 313, 315, 326, 330, 332, 333, 338, 340, 341, 344, 345, 347, 358, 362, 364

Offset: 1

Clark Kimberling, Feb 07 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*. See A299229 for a guide to related sequences.

This sequence together with A299240 and A299242 partition the positive integers.

			The first six terms are the ranks of these towers: t(4) = (2,3), t(5) = (3,2), t(18) = (3,3,2,2), t(20) = (3,2,2,3), t(22) = (3,2,3,2), t(23) = (2,3,2,3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299240, A299242.

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]
Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &]; (* this sequence *)
Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];   (* A299242 *)

A299242 Ranks of {2,3}-power towers in which #2's < #3's; see Comments.

Original entry on oeis.org

2, 7, 11, 12, 15, 16, 24, 26, 32, 33, 34, 38, 42, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 78, 86, 94, 98, 100, 101, 102, 106, 108, 109, 110, 126, 130, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 150, 154, 156, 157, 158, 166, 170, 172

Offset: 1

Clark Kimberling, Feb 07 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*. See A299229 for a guide to related sequences.

This sequence together with A299240 and A299241 partition the positive integers.

			The first six terms are the ranks of these towers: t(2) = (3), t(7) = (3,3), t(11) = (3,2,3), t(12) = (3,3,2), t(15) = (2,3,3), t(16) = (3,3,3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299240, A299241.

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]
Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &];  (* A299241 *)
Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];  (* this sequence *)

A299231 Ranks of {2,3}-power towers that start with 2; see Comments.

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

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*. See A299229 for a guide to related sequences.

			t(97) = (2,3,2,3,2,3), so that 97 is in the sequence.

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

Cf. A299229, A299232 (complement).

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]
Select[Range[200], First[t[#]] == 2 &]; (* A299231 *)
Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)

a(n) = 2n-1 for all n except 3, 4, and 6.

A299232 Ranks of {2,3}-power towers that start with 3; see Comments.

Original entry on oeis.org

2, 5, 7, 8, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124

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*. See A299229 for a guide to related sequences.

			t(76) = (3,2,3,3,2,2), so that 76 is in the sequence.

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

Cf. A299229, A299231 (complement).

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]
Select[Range[200], First[t[#]] == 2 &]; (* A299231 *)
Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)

a(n) = 2n for all n except 2, 3, and 5.

A299233 Ranks of {2,3}-power towers that end with 2; see Comments.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 14, 17, 18, 21, 22, 25, 26, 27, 28, 29, 30, 35, 36, 37, 38, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 71, 72, 73, 74, 75, 76, 77, 78, 87, 88, 89, 90, 91, 92, 93, 94, 103, 104, 105, 106, 107, 108, 109, 110, 111

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*. See A299229 for a guide to related sequences.

			t(76) = (3,2,3,3,2,2), so that 76 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299234 (complement).

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]
Select[Range[200], Last[t[#]] == 2 &]; (* A299233 *)
Select[Range[200], Last[t[#]] == 3 &]; (* A299234 *)

A299234 Ranks of {2,3}-power towers that end with 3; see Comments.

Original entry on oeis.org

2, 4, 7, 9, 11, 15, 16, 19, 20, 23, 24, 31, 32, 33, 34, 39, 40, 41, 42, 47, 48, 49, 50, 63, 64, 65, 66, 67, 68, 69, 70, 79, 80, 81, 82, 83, 84, 85, 86, 95, 96, 97, 98, 99, 100, 101, 102, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140

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*. See A299229 for a guide to related sequences.

			t(80) = (3,2,2,2,2,3), so that 80 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299233 (complement).

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]
Select[Range[200], Last[t[#]] == 2 &]; (* A299233 *)
Select[Range[200], Last[t[#]] == 3 &]; (* A299234 *)

A299235 Number of 2's in the n-th {2,3}-power tower; see Comments.

Original entry on oeis.org

1, 0, 2, 1, 1, 3, 0, 2, 2, 2, 1, 1, 4, 3, 1, 0, 3, 2, 3, 2, 3, 2, 2, 1, 2, 1, 5, 4, 4, 3, 2, 1, 1, 0, 4, 3, 3, 2, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 3, 2, 2, 1, 2, 1, 1, 0, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 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*. See A299229 for a guide to related sequences.

Every nonnegative integer occurs infinitely many times in the sequence. In particular, a(n) = 0 when the tower consists exclusively of 3's. The position of the n-th 0 in the sequence is the rank of the n-th {3}-power tower, given by 9*2^(n-2)-2 for n > 1.

			t(80) = (3,2,2,2,2,3), so that a(80) = 4.

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

Cf. A299229, A299236 (complement).

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]
Table[Count[t[n], 2], {n, 1, 100}];  (* A299235 *)
Table[Count[t[n], 3], {n, 1, 100}];  (* A299236 *)

A299236 Number of 3's in the n-th {2,3}-power tower; see Comments.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 2, 3, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 0, 1, 1, 2, 2, 3, 3, 4, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4

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*. See A299229 for a guide to related sequences.

Every nonnegative integer occurs infinitely many times in the sequence. In particular, a(n) = 0 when the tower consists exclusively of 2's. The position of the n-th 0 in the sequence is the rank of the n-th {2}-power tower, given by 7*2^(n-3) - 1 for n > 2.

			t(13) = (2,2,2,2), so that a(13) = 0.

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

Cf. A299229, A299235 (complement).

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]
Table[Count[t[n], 2], {n, 1, 100}];  (* A299235 *)
Table[Count[t[n], 3], {n, 1, 100}];  (* A299236 *)

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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A299240 Ranks of {2,3}-power towers in which #2's > #3's; see Comments.

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A299241 Ranks of {2,3}-power towers in which #2's = #3's; see Comments.

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A299242 Ranks of {2,3}-power towers in which #2's < #3's; see Comments.

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A299231 Ranks of {2,3}-power towers that start with 2; see Comments.

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A299232 Ranks of {2,3}-power towers that start with 3; see Comments.

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A299233 Ranks of {2,3}-power towers that end with 2; see Comments.

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A299234 Ranks of {2,3}-power towers that end with 3; see Comments.

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