| a+b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| 2 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 4 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 5 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 6 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
By antidiagonals, this table is sequence A3056.
The rows and columns are all A0027 (omitting one or more initial terms).
The main diagonal is A5843.
The superdiagonal and subdiagonal are both A5408.
| a×b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 2 | 0 | 2 | 4 | 6 | 8 | 10 | 12 | |
| 3 | 0 | 3 | 6 | 9 | 12 | 15 | 18 | |
| 4 | 0 | 4 | 8 | 12 | 16 | 20 | 24 | |
| 5 | 0 | 5 | 10 | 15 | 20 | 25 | 30 | |
| 6 | 0 | 6 | 12 | 18 | 24 | 30 | 36 |
By antidiagonals, this table is sequence A4247.
The rows are A0004, A0027, A5843, A8585, A8586, A008587, A8588, etc.
Each column is the same as the corresponding row.
The main diagonal is A0290.
The superdiagonal and subdiagonal are both A2378.
| ab | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 2 | 1 | 2 | 4 | 8 | 16 | 32 | 64 | |
| 3 | 1 | 3 | 9 | 27 | 81 | 243 | 729 | |
| 4 | 1 | 4 | 16 | 64 | 256 | 1024 | 4096 | |
| 5 | 1 | 5 | 25 | 125 | 625 | 3125 | 15625 | |
| 6 | 1 | 6 | 36 | 216 | 1296 | 7776 | 46656 |
By antidiagonals, this table is sequence A3992.
The rows are A0007, A0012, A0079, A0244, A0302, A0351, A0400, etc.
The columns are A0012, A0027, A0290, A0578, A0583, A0584, A1014, etc.
The main diagonal is A0312.
The superdiagonal is A7778 and the subdiagonal is A0169.
There are two ways to extend the series to a fourth operator beyond exponentiation, depending on how you group the terms. Using "^^" to represent the new operation, we can have either:
a^^b = ((a^a)^a)^...^a
or:
a^^b = a^...^(a^(a^a))
Both versions produce the same answer for a^^1 and a^^2 (a and aa respectively) but they differ beyond that. The first one produces smaller values so I call it the "lower hyper4 operator"; the other is the "higher" hyper4 operator. To distinguish between the two I use a④b and a④b respectively.
Here is a table of the lower hyper4 function for arguments from 0 to 6:
| a④b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 2 | √2 | 2 | 4 | 16 | 256 | 65536 | 4294967296 | |
| 3 | 3√3 | 3 | 27 | 19683 | 7625597484987 | 4.434265×1038 | 8.718964×10115 | |
| 4 | √2 | 4 | 256 | 4294967296 | 3.402824×1038 | 1.340781×10154 | 3.231701×10616 | |
| 5 | 5√5 | 5 | 3125 | 2.980232×1017 | 2.350989×1087 | 7.182121×10436 | 1.911013×102184 | |
| 6 | 6√6 | 6 | 46656 | 1.031442×1028 | 1.204121×10168 | 3.048039×101008 | 8.019051×106050 |
By antidiagonals, this table is sequence A171881 (more terms here; not yet in OEIS, but see here).
The first row is not in OEIS; second is A0012. Ignoring the 0 column, the next four rows are A1146, A55777, A137840, and A137841. The following rows are not in OEIS.
The first column is not an integer sequence. The next three columns are A0027, A0312, and A2489; the following columns are not in OEIS.
The main diagonal (without the initial term) is A89210.
The superdiagonal is A2488, and the subdiagonal is not in OEIS.
a④b can be generalized easily to real and even complex arguments through the identity a④b=aa(b-1).
| a④b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 2 | 1 | 2 | 4 | 16 | 65536 | 2.003530×1019728 | 106.0312×1019727 | |
| 3 | 1 | 3 | 27 | 7625597484987 | 1.26×103638334640024 | 106.0×103638334640023 | 10106.0×103638334640023 | |
| 4 | 1 | 4 | 256 | 1.340781×10154 | 108.0723×10153 | 10108.0723×10153 | 3 PT 8.0723×10153 | |
| 5 | 1 | 5 | 3125 | 1.911013×102184 | 101.335740×102184 | 10101.3357×102184 | 3 PT 1.3357×102184 | |
| 6 | 1 | 6 | 46656 | 2.659120×1036305 | 102.0692×1036305 | 10102.0692×1036305 | 3 PT 2.0692×1036305 |
By antidiagonals, this table is sequence A171882 (more terms here; not yet in OEIS, but see here).
First four rows are A0035, A0012, A14221, and A14222; following rows are not in OEIS.
First four columns are A0012, A0027, A0312, and A002488; following columns are not in OEIS.
The higher hyper4 operator cannot be extended in any obvious and self-consistent way to the reals (to compute, for example, {pi}④e). This issue is discussed at length here, with possible solutions.
The hyper4 function is also discussed extensively on my large numbers page.
A171881: 0, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 27, 16, 1, 1, 5, 256, 19683, 256, 1, 1, 6, 3125, 4294967296, 7625597484987, 65536, 1, 1, 7, 46656, 298023223876953125, 340282366920938463463374607431768211456, 443426488243037769948249630619149892803, 4294967296, 1, 1, 8, 823543, ...
A171882: 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 4, 1, 1, 1, 4, 27, 16, 1,
0, 1, 5, 256, 7625597484987, 65536, 1, 1, 1, 6, 3125,
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096,
(next term is 3^7625597484987~=1.26*10^3638334640024), ...
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