cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

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

Extensions

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

A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 851, 1928, 4396, 10087, 23273, 53948, 125608, 293543, 688366, 1619087, 3818818, 9029719, 21400706, 50828664, 120963298, 288405081, 688821573, 1647853491, 3948189131, 9473431479
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) <= A002955(n). - Max Alekseyev, Sep 23 2009

Examples

			From _M. F. Hasler_, Apr 17 2024: (Start)
The table with explicit lists of values starts as follows:
   n | distinct values of 2^...^2 with all possible parenthesizations
-----+---------------------------------------------------------------
   1 | 2
   2 | 2^2 = 4
   3 | (2^2)^2 = 2^(2^2) = 16
   4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
   5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
   6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
   7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
     | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
  ...| ...
(When parentheses are omitted above, we use that ^ is right associative.) (End)

References

  • J. Q. Longyear, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

Programs

  • PARI
    /* Define operators for numbers represented (recursively) as list of positions of bits 1. Illustration using the commands below: T = 3.bits; T.int */
n.bits = vector(hammingweight(n), v,  n -= 1 << v= valuation(n, 2); v.bits)
ONE = 1.bits; m.int = sum(i=1, #m, 1<=0])}
{ADD(m, n, a=#m, b=#n)= if(!a, n, !b, m, a=b=1; until(a>#m|| b>#n, if(m[a]==n[b], until(a>=#m|| m[a]!=m[a+1]|| !#m=m[^a], m[a]=ADD(m[a],ONE)); b++, CMP(m[a], n[b])<0, a++, m=concat([m[1..a-1], [n[b]], m[a..#m]]); b++)); b>#n|| m=concat(m,n[b..#n]); m)}
{CMP(m, n, a=#m, b=#n, c=0)= if(!b, a, !a, -1, while(!(c=CMP(m[a], n[b]))&& a--&& b--, ); if(c, c, 1-b))}
{SUB(m, n, a=#n)= if(!a, m, my(b=a=1, c, i); while(a<=#m && b<=#n, if(0>c=CMP(m[a], n[b]), a++, c, i=[c=n[b]]; b++; while(m[a]!=c=ADD(c, ONE), if(b<=#n && c==n[b], b++, i=concat(i, [c]))); m=concat([m[1..a-1], i, m[a+1..#m]]); a += #i, m=m[^a]; b++)); m)}
A2845 = List([[2.bits]]) /* List of values for each n */
{A002845(n)= while(#A2845= 15. - M. F. Hasler, Apr 28 2024

Extensions

a(12)-a(13) corrected and a(14)-a(27) added by Jon E. Schoenfield, Oct 11 2008
a(28)-a(29) computed by Kirill Osenkov, added by Vladimir Reshetnikov, Feb 07 2019
a(30)-a(31) added by Sean A. Irvine, Feb 18 2019

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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]

References

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

Links

Crossrefs

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

Extensions

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

Views

Author

Jon E. Schoenfield, Oct 13 2008

Keywords

Links

Crossrefs

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

Views

Author

Jon E. Schoenfield, Oct 13 2008

Keywords

Links

Crossrefs

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

Views

Author

Jon E. Schoenfield, Oct 13 2008

Keywords

Links

Crossrefs

Cf. A002845, A003018, A003019, A145545, A145546, A145548, 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

Views

Author

Jon E. Schoenfield, Oct 13 2008

Keywords

Comments

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

Links

Crossrefs

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

Views

Author

Jon E. Schoenfield, Oct 13 2008

Keywords

Links

Crossrefs

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

A199205 Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683, 11524, 12393
Offset: 1

Views

Author

Alois P. Heinz, Nov 03 2011

Keywords

Examples

			a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.

Crossrefs

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.

Programs

  • Maple
    f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0]},
                {seq(seq(seq( [2+g[1], 3*(1 +g[1] +h[1]) +g[2],
                 8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3]],
                 h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);
  • Mathematica
    f[n_] := f[n] = If[n == 1, {{0, 0, 0}}, Union @ Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(41)-a(42) from Alois P. Heinz, Jun 01 2015

A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 92, 182, 342, 601, 982, 1499, 2169, 2970, 3994, 5297, 6834, 8635, 10714, 13121, 16104, 19674, 23868, 28453, 33637, 39630, 46730
Offset: 1

Views

Author

Alois P. Heinz, Nov 04 2011

Keywords

Examples

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.

Crossrefs

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215835. Column k=5 of A216368.

Programs

  • Maple
    f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0, 0]},
            {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
             8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
             10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
             +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
              h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[4], f(n))):
seq(a(n), n=1..20);
  • Mathematica
    f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length@Union@(#[[4]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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