A002845 -id:A002845 - cp's OEIS frontend

A072925 Probably an erroneous version of A002845.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 888, 1944

Offset: 1

dead

The old entry with this sequence number was a duplicate of A037444.

J. Q. Longyear, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995; entry M1139.

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
J. Longyear, T. Trotter, N. J. A. Sloane, Correspondence

A003018 Number of distinct values taken by 3^3^...^3 (with n 3's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 111, 270, 664, 1659, 4184, 10662, 27367, 70747, 183925, 480656, 1261630, 3324772, 8792592, 23327249, 62067785, 165586565

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
Index entries for sequences related to parenthesizing
MathOverflow discussion of related questions

Cf. A002845, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

a(12)-a(23) from Jon E. Schoenfield, Oct 11 2008

A003019 Number of distinct values taken by 4^4^...^4 (with n 4's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 114, 282, 703, 1787, 4583, 11900, 31131, 82117, 217954, 581970, 1561704, 4210263, 11396488, 30963024, 84402984, 230779071, 632762424, 1739387089

Offset: 1

N. J. A. Sloane

See also the Four Fours puzzle [Bourke]. Four fours is a mathematical puzzle. The goal of four fours is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four (no other digit is allowed). The subsequence of primes begins 2, 1787, 4583, no more through a(23). [Jonathan Vos Post, Apr 02 2011]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Paul Bourke, Four Fours Problem.
Index entries for sequences related to parenthesizing
MathOverflow discussion of related questions

Cf. A002845, A003018, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

a(12)-a(23) from Jon E. Schoenfield, Oct 11 2008

a(24)-a(25) from Marek Hubal, Mar 01 2019

A145545 Number of distinct values taken by 5^5^...^5 (with n 5's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 285, 715, 1826, 4711, 12303, 32387, 85964, 229662, 617300, 1667787, 4527290, 12340688, 33766612, 92707834, 255329106

Offset: 1

Jon E. Schoenfield, Oct 13 2008

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
MathOverflow, MathOverflow discussion of related questions
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145546, A145547, A145548, A145549, A145550, A000081.

A145546 Number of distinct values taken by 6^6^...^6 (with n 6's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1838, 4750, 12431, 32790, 87225, 233534, 629123, 1703586, 4635181, 12664335, 34734322, 95592704, 263909594

Offset: 1

Jon E. Schoenfield, Oct 13 2008

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145545, A145547, A145548, A145549, A145550, A000081.

A145547 Number of distinct values taken by 7^7^...^7 (with n 7's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4762, 12470, 32918, 87628, 234795, 633000, 1715435, 4671098, 12772707, 35059815, 96567161, 266818396, 739344427

Offset: 1

Jon E. Schoenfield, Oct 13 2008

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145545, A145546, A145548, A145549, A145550, A000081.

A145548 Number of distinct values taken by 8^8^...^8 (with n 8's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4765, 12482, 32957, 87756, 235198, 634261, 1719312, 4682952, 12808650, 35168306, 96893138, 267794711, 742260014, 2062792103

Offset: 1

Jon E. Schoenfield, Oct 13 2008

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145545, A145546, A145547, A145549, A145550, A000081.

A145549 Number of distinct values taken by 9^9^...^9 (with n 9's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12485, 32969, 87795, 235326, 634664, 1720573, 4686829, 12820504, 35204254, 97001655, 268120807, 743236814, 2065709551, 5755253457

Offset: 1

Jon E. Schoenfield, Oct 13 2008

The subsequence of primes begins 2, 719, 32969. No more through a(26). [Jonathan Vos Post, Apr 02 2011]

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Jonathan Vos Post, The Four Nines problem.
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145550, A000081.

A145550 Number of distinct values taken by 10^10^...^10 (with n 10's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32972, 87807, 235365, 634792, 1720976, 4688090, 12824381, 35216108, 97037603, 268229329, 743562936, 2066686470, 5758171390, 16079351152

Offset: 1

Jon E. Schoenfield, Oct 13 2008

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Index entries for sequences related to parenthesizing

Cf. A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A000081.

A082499 Take a string of n x's and insert n-1 ^'s and n-2 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2).

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 38, 88, 206, 497, 1212, 2996

Offset: 1

W. Edwin Clark and Wouter Meeussen, Apr 29 2003

For n=10, largest value is 2^(2^127) = x^(x^(x^(x^(x^6)))) = x^(x^(x^((((((x^x)^x)^x)^x)^x)^x))) and results from the 132nd tree {0,{0,{0,{{{{{{0,0},0},0},0},0},0}}}} or {1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0}.

			For n = 4 there are 5 functions: f1(x) = ((x^x)^x)^x; f2(x) = (x^(x^x))^x; f3(x) = x^((x^x)^x); f4(x) = x^(x^(x^x)); f5(x) = (x^x)^(x^x); but only 4 different values when x = sqrt(2).

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
Index entries for sequences related to parenthesizing

Cf. A003019, A000081, A002845, A003018, A082543.

trees[1] = {x};
trees[n_] := trees[n] = Flatten@Table[ch1^ch2, {k, n-1}, {ch1, trees[k]}, {ch2, trees[n-k]}];
logs[t_] := ((log/@t) //. {log[a_^b_]:>log[a]b, log[a_ b_]:>log[a]+log[b], log[x]->one, log[one]->0});
Table[Length@Union[logs@logs@trees[n] /. {one->1, x->Sqrt[2]}, SameTest->Equal], {n, 9}] (* Andrei Zabolotskii, Jan 03 2025 *)

Term a(11) = 1212 added by Vladimir Reshetnikov, Oct 29 2011
a(1) added by Franklin T. Adams-Watters, Nov 03 2011
a(12) from Andrei Zabolotskii, Jul 23 2025

cp's OEIS Frontend

A072925 Probably an erroneous version of A002845.

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A003018 Number of distinct values taken by 3^3^...^3 (with n 3's and parentheses inserted in all possible ways).

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A003019 Number of distinct values taken by 4^4^...^4 (with n 4's and parentheses inserted in all possible ways).

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A145545 Number of distinct values taken by 5^5^...^5 (with n 5's and parentheses inserted in all possible ways).

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A145546 Number of distinct values taken by 6^6^...^6 (with n 6's and parentheses inserted in all possible ways).

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A145547 Number of distinct values taken by 7^7^...^7 (with n 7's and parentheses inserted in all possible ways).

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A145548 Number of distinct values taken by 8^8^...^8 (with n 8's and parentheses inserted in all possible ways).

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A145549 Number of distinct values taken by 9^9^...^9 (with n 9's and parentheses inserted in all possible ways).

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A145550 Number of distinct values taken by 10^10^...^10 (with n 10's and parentheses inserted in all possible ways).

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A082499 Take a string of n x's and insert n-1 ^'s and n-2 pairs of parentheses in all possible legal ways. Sequence gives number of distinct values when x = sqrt(2).

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