A256131 -id:A256131 - cp's OEIS frontend

A054871 a(n) = H_n(3,2) where H_n is the n-th hyperoperator.

3, 5, 6, 9, 27, 7625597484987

Offset: 0

Walter Nissen, May 28 2000

nonn

H_n(x,y) is defined recursively by:
  H_0(x,y) = y+1;
  H_1(x,0) = x;
  H_2(x,0) = 0;
  H_n(x,0) = 1, for n>2;
  H_n(x,y) = H_{n-1}(x,H_n(x,y-1)), for integers n>0 and y>0.
Consequently:
  H_0(x,y) = y+1  is the successor function on y;
  H_1(x,y) = x+y  is addition;
  H_2(x,y) = x*y  is multiplication;
  H_3(x,y) = x^y  is exponentiation;
  H_4(x,y) = x^^y is tetration (a height-y exponential tower x^x^x^... );
  ...
Extending to negative-order hyperoperators via the recursive formula:
  H_0(x,y) = H_{-1}(x,H_0(x,y-1)) = H_{-1}(x,y).
Therefore:
  H_{-n}(x,y) = H_0(x,y), for every nonnegative n.
This function is an Ackermann function variant because it satisfies the recurrence relation above (see A046859).
Other hyperoperation notations equivalent to H_n(x,y) include:
Square Bracket or Box: a [n] b;
Conway Chain Arrows: a -> b -> n-2;
Knuth Up-arrow: a "up-arrow"(n-2) b;
Standard Caret: a ^(n-2) b.
Originally published as 3 agg-op-n 3 for n > 0. - Natan Arie Consigli, Apr 22 2015
Sequence can also be defined as a(0) = 3, a(1) = 5, a(n) = H_{n-1}(3,3) for n > 1. - Natan Arie Consigli, Apr 22 2015; edited by Danny Rorabaugh, Oct 18 2015
Before introducing the H_n notation, this sequence was named "3 agg-op-n 2, where the binary aggregation operators agg-op-n are zeration, addition, multiplication, exponentiation, superexponentiation, ..." - Danny Rorabaugh, Oct 14 2015
The next term is 3^3^...^3 (with 7625594784987 3's). - Jianing Song, Dec 25 2018

			a(0) = H_0(3,2) = 2+1 = 3;
a(1) = H_1(3,2) = 3+2 = 5;
a(2) = H_2(3,2) = 3*2   = 3+3  = 6;
a(3) = H_3(3,2) = 3^2   = 3*3  = 9;
a(4) = H_4(3,2) = 3^^2  = 3^3  = 27;
a(5) = H_5(3,2) = 3^^^2 = 3^^3 = 3^(3^3) = 7625597484987.

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 60.

Rick Norwood, Math. Bite: Why 2 + 2 = 2 * 2, Mathematics Magazine, Vol. 71 (1998), p. 60.
Stephen R. Wassell, Superexponentiation and Fixed Points of Exponential and Logarithmic Functions, Mathematics Magazine, Vol. 73 (2000), pp. 111-119.
Eric Weisstein's MathWorld, Ackermann Function and Power Tower
Wikipedia, Hyperoperation
Index Section Ho-Hy

H_n(x,y) for various x,y: A001695 (2,n), this sequence (3,2; almost 3,3), A067652 (2,3; almost 2,4), A141044 (1,1), A175796 (n,2), A179184 (0,0), A189896 (n,n), A213619 (n,H_n(n,n)), A253855 (4,2; almost 4,4), A255176 (2,2),  A255340 (4,3), A256131 (10,2; almost 10,10), A261143 (1,2), A261146 (n,3). - Natan Arie Consigli and Danny Rorabaugh, Oct 14-26 2015
H_4(x,n) for various x: A000035 (x=0), A014221 (x=2), A014222 (x=3, shifted), A057427 (x=1).
H_5(x,n) for various x: A266198 (x=2), A266199 (x=3).
Cf. A254225, A254310, A257229.

First two terms prepended by Natan Arie Consigli, Apr 22 2015
First term corrected and hyperoperator notation implemented by Danny Rorabaugh, Oct 14 2015
Definition extended to include negative n by Natan Arie Consigli, Oct 19 2015
More hyperoperator notation added by Natan Arie Consigli, Jan 19 2016

A189896 Weak Ackermann numbers: H_n(n,n) where H_n is the n-th hyperoperator.

Original entry on oeis.org

1, 2, 4, 27

Offset: 0

Max Sills, Apr 30 2011

The next term, a(4), has about 8*10^153 decimal digits. - Charles R Greathouse IV, Nov 15 2022

			a(0) = succ(0) = 0 + 1 = 1, because the zeroth hyperoperation is successor.
a(1) = 1 + 1 = 2, because the first hyperoperation is addition.
a(2) = 2 * 2 = 4, because the second hyperoperation is multiplication.
a(3) = 3^3 = 27, because the third hyperoperation is exponentiation.
a(4) = 4^4^4^4 = 4^(4^(4^4)) = 4^(4^256), because the fourth hyperoperation is tetration. The term is too big to be included: log_2(a(4)) = 2^513.

M. H. Löb and S. S. Wainer, Hierarchies of number-theoretic functions. I, Archive for Mathematical Logic 13:1-2 (1970), pp. 39-51.
Wikipedia, Hyperoperation.
Wikipedia, Ackermann function

For H_n(x,x) with fixed x, cf. A054871 (x=3, shifted), A141044 (x=1), A253855 (x=4, shifted), A255176 (x=2), A256131 (x=10, shifted). - Danny Rorabaugh, Oct 20 2015

Cf. A271553 ( H_n-1(n,n) ). - Natan Arie Consigli, Apr 10 2016

a(n) = H_n(n, n), where H_n the hyperoperation indexed by n.

"Weak" added to definition by Natan Arie Consigli, Apr 18 2015

A175796 H_n(n, 2) where H_c(a, b) is the hyperoperation function with operator c.

Original entry on oeis.org

3, 3, 4, 9, 256

Offset: 0

Grant Garcia, Sep 06 2010

nonn

			a(0) = H_0(0, 2) = 2 + 1 = 3
a(1) = H_1(1, 2) = 1 + 2 = 3
a(2) = H_2(2, 2) = 2 * 2 = 4
a(3) = H_3(3, 2) = 3 ^ 2 = 9
a(4) = H_4(4, 2) = 4 ^^ 2 = 4 ^ 4 = 256
a(5) = H_5(5, 2) = 5 ^^^ 2 = 5 ^^ 5 = 5 ^ 5 ^ 5 ^ 5 ^ 5 =~ 10 ^ (10 ^ (10 ^ (2184.1257...)))

Wikipedia, Hyperoperation
Index Section Ho-Hy

For H_n(x,2) with fixed x, cf. A054871 (x=3), A253855 (x=4), A255176 (x=2), A256131 (x=10), A261143 (x=1). - Danny Rorabaugh, Oct 20 2015

def H(a, b, c):
    if c == 0: return b + 1
    if c == 1 and b == 0: return a
    if c == 2 and b == 0: return 0
    if c >= 3 and b == 0: return 1
    return H(a, H(a, b - 1, c), c - 1)
for n in range(5): print(H(n, 2, n))

a(n) = H_n(n, 2)

H_c(a, b) = {b + 1 if c = 0; a if c = 1, b = 0; 0 if c = 2, b = 0; 1 if c >= 3, b = 0; H_{c-1}(a, H_c(a, b - 1)) otherwise}

A280265 Array with five columns read by rows: H_k(n,2), with rows n >= 0 and columns 0 <= k <= 4, where H_n is the n-th hyperoperation.

Original entry on oeis.org

3, 2, 0, 1, 1, 3, 3, 2, 1, 1, 3, 4, 4, 4, 4, 3, 5, 6, 9, 27, 3, 6, 8, 16, 256, 3, 7, 10, 25, 3125, 3, 8, 12, 36, 46656, 3, 9, 14, 49, 823543, 3, 10, 16, 64, 16777216, 3, 11, 18, 81, 387420489, 3, 12, 20, 100, 10000000000, 3, 13, 22, 121, 285311670611, 3, 14, 24, 144, 8916100448256

Offset: 0

Natan Arie Consigli, Dec 30 2016

See A054871 for definitions and key links.

The purpose of this sequence is to unify all the visible terms of the sequence: a(k)= H_k(n,2) for some n.

			Square array begins:
3, 2,  0,  1,   1;
3, 3,  2,  1,   1;
3, 4,  4,  4,   4;
3, 5,  6,  9,   27;
3, 6,  8,  16,  256;
3, 7,  10, 25,  3125;
3, 8,  12, 36,  46656;
3, 9,  14, 49,  823543;
3, 10, 16, 64,  16777216;
3, 11, 18, 81,  387420489;
3, 12, 20, 100, 10000000000;
3, 13, 22, 121, 285311670611;
3, 14, 24, 144, 8916100448256;
...
For row 10 we have:
H_0(10,2) = 3;
H_1(10,2) = 12;
H_2(10,2) = 20;
H_3(10,2) = 100;
H_4(10,2) = 10000000000;

Cf. A054871, A256131 (contains only line n=10), A280267, A000312.

H[0, x_, y_] := y + 1;
H[1, x_, y_] := x + y;
H[2, x_, y_] := x*y;
H[3, x_, y_] := x^y;
H[4, x_, 2] := x^x;
Table[H[k, n, 2], {n, 0, 20}, {k, 0, 4}]

Definition corrected by Natan Arie Consigli, Jun 13 2017

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A054871 a(n) = H_n(3,2) where H_n is the n-th hyperoperator.

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A189896 Weak Ackermann numbers: H_n(n,n) where H_n is the n-th hyperoperator.

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A175796 H_n(n, 2) where H_c(a, b) is the hyperoperation function with operator c.

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A280265 Array with five columns read by rows: H_k(n,2), with rows n >= 0 and columns 0 <= k <= 4, where H_n is the n-th hyperoperation.

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