This page is for types of numbers I have come up with (or at least thought of, not sure if it's already been claimed)
A pyraginal (peer-ag-in-al) number is a number that is able to be represented as n^n, and contains at least one "n" within it's digits.
PN(1) = 1^1 = 1
PN(2) = 5^5 = 3,125
PN(3) = 6^6 = 46,656
PN(4) = 9^9 = 38,420,489
PN(5) = 10^10 = 10,000,000,000
PN(6) = 11^11 = 285,311,670,611
PN(7) = 16^16 = 18,446,744,073,709,551,616
PN(8) = 17^17 = 827,240,261,886,336,764,177
PN(9) = 19^19 = 1,978,419,655,660,313,589,123,979
PN(10) = 21^21 = 5,842,587,018,385,982,521,381,124,421
and so on.
Double-pyraginal numbers are numbers that can be represented as n^n and contain at least two "n"s within their digits.
DPN(1) = 6^6 = 46,656
DPN(2) = 11^11 = 285,311,670,611
DPN(3) = 16^16 = 18,446,744,073,709,551,616
DPN(4) = 19^19 = 1,978,419,655,660,313,589,123,979
DPN(5) = 21^21 = 5,842,587,018,385,982,521,381,124,421
DPN(6) = 25^25 = 88,817,841,970,012,523,233,890,533,447,265,625
and so on.
Triple-pyraginal numbers are numbers that can be written as n^n, and contain at least three "n"s within their digits.
TPN(1) = 6^6 = 46,656
TPN(2) = 35^35 = 1,102,507,499,354,148,695,951,786,433,413,508,348,166,942,596,435,546,875
TPN(3) = 45^45 = 248,063,644,451,341,145,494,649,182,395,412,689,744,530,581,492,654,164,321,720,600,128,173,828,125
and so on. You can guess what Quadruple-Pyraginal, Quintuple-Pyraginal, and so on would be. In fact, m-Pyraginal Numbers is something, meaning for m-Pyraginal, there must be m amount of "n"s in the digits.
A factagonal number is a number that can be represented by n!, and contains at least one "n" within it's digits.
FN(1) = 1! = 1
FN(2) = 2! = 2
FN(3) = 4! = 24
FN(4) = 20! = 2,432,902,008,176,640,000
FN(5) = 21! = 51,090,942,171,709,440,000
FN(6) = 26! = 403,291,461,126,605,635,584,000,000
FN(7) = 30! = 265,252,859,812,191,058,636,308,480,000,000
FN(8) = 37! = 13,763,753,091,226,345,046,315,979,581,580,902,400,000,000
FN(9) = 39! = 20,397,882,081,197,443,358,640,281,739,902,897,356,800,000,000
FN(10) = 42! = 1,405,006,117,752,879,898,543,142,606,244,511,569,936,384,000,000,000
and so on. You can guess what Double-Factagonal, Triple-Factagonal, and so on are. There are in fact infinite Factagonal functions, expanding similarly to the Pyraginal functions.
DFN(1), or the first Double Factagonal Number, is 60!.
DFN(1) = 8,320,987,112,741,390,144,276,341,183,223,364,380,754,172,606,361,245,952,449,277,696,409,600,000,000,000,000
This is a number type Googology Wiki user AlexJN developed as a solution for a potential issue for Factagonal numbers simply changing by (n+1)!Â
It is defined by AlexJN as so:
Fx(1)=1!=1
Fx(n) when n>1 is the smallest factorial which has Fx(n-1) contained in it
Fx(2)=5!=120
Fx(3)=14!=87,178,291,200
As you can see, this system generates far larger numbers than FN(n).
I would like to expand this in the same way the others expand, each subsequent function requiring one more of the number to be found within.
1Fx(n)
2Fx(n)
3Fx(n)
and so forth. mFx(1) = 1 always.
HG(n). There are 6+(n^2) dots, 6 of which are shaped in a hexagon, with the rest dispersed evenly within. Make a sequence of graphs, notated as g(1, 1), g(1, 2), g(1, y), g(2, y), g(x, y)... connect the dots with x lines, no more than n+1 lines connecting one dot. y is the number of the graph, and the output of HG(n). No More than x lines per graph. A shape is any lines that are connected to each other. Once a shape is used, that exact shape cannot be used in any other graphs. Once all possible graphs are made with x lines, and not reusing any shapes, change x by 1. No matter the rotation or reflection of the shape, it cannot be used again. y changes by 1 for every new, valid graph. In the end, the y value is the total number of graphs. HG(n) = the y value.
Multi-Entry Extension
HG(n, b) = HG^b(n) = HG(HG(HG(...HG(HG(n))...))) with b HG()'s
HG(n, b, 2) = HG(n, HG(n, HG(...HG(n, HG(n, b))...))) with b recursions
HG(n, b, 3) = HG(n, HG(n, HG(...HG(n, HG(n, b, 2), 2)...), 2), 2) with b recursions
HG(n, b, c) = HG(n, HG(n, HG(...HG(n, HG(n, b, c-1), c-1)...), c-1), c-1) with b recursions
HG(n, b, c, 2) = HG(n, HG(n, HG(...HG(n, HG(n, b, c), c)...), c), c) with b recursions
HG(n, b, c, d) = HG(n, HG(n, HG(n, HG(...HG(n, HG(n, b, c, d-1), c, d-1)...), c, d-1)), c, d-1) with b recursions
and so on, with any number of entries
HG([n]) = HG(n, n, n...n, n, n) with n n's
.....................................................................
I believe HG(1) = 65
Continuing the extension
HG([n]2) = HG([HG([HG([...HG([HG([n])])...])])]) with n HG([])'s
HG([n]3) = HG([HG([HG([...HG([HG([n]2)]2)...]2)]2)]2) with n HG([]2)'s
HG([n]b) = HG([HG([HG([...HG([HG([n]b-1)]b-1)...]b-1)]b-1)]b-1) with n HG([]b-1)'s
HG([n]1,2) = HG([HG([HG([...HG([HG([n]n)]n)...]n)]n)]n) with n nestings
HG([n]b,2) = HG([HG([HG([...HG([HG([n]b-1, 2)]b-1, 2)...]b-1,2)]b-1,2)]b-1,2) with n nestings
HG([n]1,3) = HG([HG([HG([...HG([HG([n]n, 2)]n,2)...]n,2)]n,2)]n,2) with n nestings
HG([n]b,3) = HG([HG([HG([...HG([HG([n]b-1,3)]b-1,3)...]b-1,3)]b-1,3)]b-1,3) with n nestings
HG([n]1,c) = HG([HG([HG([...HG([HG([n]n,c-1)]n,c-1)...]n,c-1)]n,c-1)]n,c-1) with n nestings
HG([n]b,c) = HG([HG([HG([...HG([HG([n]b-1,c)]b-1,c)...]b-1,c)]b-1,c)]b-1,c) with n nestings
HG([n]1,1,2) = HG([HG([HG([...HG([HG([n]n,n)]n,n)...]n,n)]n,n)]n,n) with n nestings
HG([n]2,1,2) = HG([HG([HG([...HG([HG([n]1,1,2)]1,1,2)...]1,1,2)]1,1,2)]1,1,2) with n nestings
HG([n]b,1,2) = HG([HG([HG([...HG([HG([n]b-1,1,2)]b-1,1,2)...]b-1,1,2)]b-1,1,2)]b-1,1,2) with n nestings
HG([n]1,2,2) = HG([HG([HG([...HG([HG([n]n,1,2)]n,1,2)...]n,1,2)]n,1,2)]n,1,2) with n nestings
HG([n]2,2,2) = HG([HG([HG([...HG([HG([n]1,2,2)]1,2,2)...]1,2,2)]1,2,2)]1,2,2) with n nestings
HG([n]1,1,3) = HG([HG([HG([...HG([HG([n]n,n,2)]n,n,2)...]n,n,2)]n,n,2)]n,n,2) with n nestings
HG([n]1,1,d) = HG([HG([HG([...HG([HG([n]n,n,d-1)]n,n,d-1)...]n,n,d-1)]n,n,d-1)]n,n,d-1) with n nestings
HG([n]b,c,d) = HG([HG([HG([...HG([HG([n]b-1,c,d)]b-1,c,d)...]b-1,c,d)]b-1,c,d)]b-1,c,d) with n nestings
and so on, with any number of entries after the [ ]
HG([n], 2) = HG([n]n, n, n...n, n, n) with n entries of n after the [ ]
HG([n], b) = HG([HG([HG([...HG([HG([n], b-1)], b-1)...], b-1)], b-1)], b-1) with n nestings
HG([n], 1, 2) = HG([HG([HG([...HG([HG([n], n)], n)...], n)], n)], n) with n nestings
HG([n], 2, 2) = HG([HG([HG([...HG([HG([n], 1, 2)], 1, 2)...], 1, 2)], 1, 2)], 1, 2) with n nestings
HG([n], b, 2) = HG([HG([HG([...HG([HG([n], b-1, 2)], b-1, 2)...], b-1, 2)], b-1, 2)], b-1, 2) with n nestings
HG([n], 1, 3) = HG([HG([HG([...HG([HG([n], n, 2)], n, 2)...], n, 2)], n, 2)], n, 2) with n nestings
HG([n], 1, c) = HG([HG([HG([...HG([HG([n], n, c-1)], n, c-1)...], n, c-1)], n, c-1)], n, c-1) with n nestings
HG([n], b, c) = HG([HG([HG([...HG([HG([n], b-1, c)], b-1, c)...], b-1, c)], b-1, c)], b-1, c) with n nestings
HG([n], 1, 1, 2) = HG([HG([HG([...HG([HG([n]n, n)]n, n)...]n, n)]n, n)]n, n) with n nestings
HG([n], 2, 1, 2) = HG([HG([HG([...HG([HG([n], 1, 1, 2)], 1, 1, 2)...], 1, 1, 2)], 1, 1, 2)], 1, 1, 2) with n nestings
HG([n], 1, 2, 2) = HG([HG([HG([...HG([HG([n]n, 1, 2)]n, 1, 2)...]n, 1, 2)]n, 1, 2)]n, 1, 2) with n nestings
HG([n], 1, 1, 3) = HG([HG([HG([...HG([HG([n], n, n, 2)], n, n, 2)...], n, n, 2)], n, n, 2)], n, n, 2) with n nestings
HG([n], b, c, d) = HG([HG([HG([...HG([HG([n], b-1, c, d)], b-1, c, d)...], b-1, c, d)], b-1, c, d)], b-1, c, d) with n nestings
and so on, with any number of entries after the [ ]
HG([n, 2]) = HG([n], n, n, n...n, n, n) with n n's after the [ ]
HG([n, b]) = HG([HG([HG([...HG([HG([n, b-1]), b-1])..., b-1]), b-1]), b-1]) with n nestings
HG([n, b]2) = HG([HG([HG([...HG([HG([n, b]), b])..., b]), b]), b]) with n HG([])'s
HG([n, b]3) = HG([HG([HG([...HG([HG([n, b]2), b]2)..., b]2), b]2), b]2) with n HG([]2)'s
HG([n, b]e) = HG([HG([HG([...HG([HG([n, bb-1), b]b-1)..., b]b-1), b]b-1), b]b-1) with n HG([]b-1)'s
HG([n, b]1,2) = HG([HG([HG([...HG([HG([n, b]n), b]n)..., b]n), b]n), b]n) with n nestings
HG([n, b]e,2) = HG([HG([HG([...HG([HG([n, b]e-1, 2), b]e-1, 2)..., b]e-1,2), b]e-1,2), b]e-1,2) with n nestings
HG([n, b]1,3) = HG([HG([HG([...HG([HG([n, b]n, 2), b]n,2)..., b]n,2), b]n,2), b]n,2) with n nestings
HG([n, b]e,3) = HG([HG([HG([...HG([HG([n, b]e-1,3), b]e-1,3)..., b]e-1,3), b]e-1,3), b]e-1,3) with n nestings
HG([n, b]1,c) = HG([HG([HG([...HG([HG([n, b]n,c-1), b]n,c-1)..., b]n,c-1), b]n,c-1), b]n,c-1) with n nestings
HG([n, b]e,c) = HG([HG([HG([...HG([HG([n, b]e-1,c), b]e-1,c)..., b]e-1,c), b]e-1,c), b]e-1,c) with n nestings
HG([n, b]1,1,2) = HG([HG([HG([...HG([HG([n, b]n,n), b]n,n)..., b]n,n), b]n,n), b]n,n) with n nestings
HG([n, b]2,1,2) = HG([HG([HG([...HG([HG([n, b]1,1,2), b]1,1,2)..., b]1,1,2), b]1,1,2), b]1,1,2) with n nestings
HG([n, b]e,1,2) = HG([HG([HG([...HG([HG([n, b]e-1,1,2), b]e-1,1,2)..., b]e-1,1,2), b]e-1,1,2), b]e-1,1,2) with n nestings
HG([n, b]1,2,2) = HG([HG([HG([...HG([HG([n, b]n,1,2), b]n,1,2)..., b]n,1,2), b]n,1,2), b]n,1,2) with n nestings
HG([n, b]2,2,2) = HG([HG([HG([...HG([HG([n, b]1,2,2), b]1,2,2)..., b]1,2,2), b]1,2,2), b]1,2,2) with n nestings
HG([n, b]1,1,3) = HG([HG([HG([...HG([HG([n, b]n,n,2), b]n,n,2)..., b]n,n,2), b]n,n,2), b]n,n,2) with n nestings
HG([n, b]1,1,d) = HG([HG([HG([...HG([HG([n, b]n,n,d-1), b]n,n,d-1)..., b]n,n,d-1), b]n,n,d-1), b]n,n,d-1) with n nestings
HG([n, b]e,c,d) = HG([HG([HG([...HG([HG([n, b]e-1,c,d), b]e-1,c,d)..., b]e-1,c,d), b]e-1,c,d), b]e-1,c,d) with n nestings
and so on, with any number of entries after the [ ]
HG([n, b], 2) = HG([n, b]n, n, n...n, n, n) with n entries of n after the [ ]
HG([n, b], e) = HG([HG([HG([...HG([HG([n, b], b-1), b], b-1)..., b], b-1), b], b-1), b], b-1) with n nestings
HG([n, b], 1, 2) = HG([HG([HG([...HG([HG([n, b], n), b], n)..., b], n), b], n), b], n) with n nestings
HG([n, b], 2, 2) = HG([HG([HG([...HG([HG([n, b], 1, 2), b], 1, 2)..., b], 1, 2), b], 1, 2), b], 1, 2) with n nestings
HG([n, b], e, 2) = HG([HG([HG([...HG([HG([n, b], b-1, 2), b], b-1, 2)..., b], b-1, 2), b], b-1, 2), b], b-1, 2) with n nestings
HG([n, b], 1, 3) = HG([HG([HG([...HG([HG([n, b], n, 2), b], n, 2)..., b], n, 2), b], n, 2), b], n, 2) with n nestings
HG([n, b], 1, c) = HG([HG([HG([...HG([HG([n, b], n, c-1), b], n, c-1)..., b], n, c-1), b], n, c-1), b], n, c-1) with n nestings
HG([n, b], e, c) = HG([HG([HG([...HG([HG([n, b], b-1, c), b], b-1, c)..., b], b-1, c), b], b-1, c), b], b-1, c) with n nestings
HG([n, b], 1, 1, 2) = HG([HG([HG([...HG([HG([n, b]n, n), b]n, n)..., b]n, n), b]n, n), b]n, n) with n nestings
HG([n, b], 2, 1, 2) = HG([HG([HG([...HG([HG([n, b], 1, 1, 2), b], 1, 1, 2)..., b], 1, 1, 2), b], 1, 1, 2), b], 1, 1, 2) with n nestings
HG([n, b], 1, 2, 2) = HG([HG([HG([...HG([HG([n, b], n, 1, 2), b]n, 1, 2)..., b]n, 1, 2), b]n, 1, 2), b]n, 1, 2) with n nestings
HG([n, b], 1, 1, 3) = HG([HG([HG([...HG([HG([n, b], n, n, 2), b], n, n, 2)..., b], n, n, 2), b], n, n, 2), b], n, n, 2) with n nestings
HG([n, b], e, c, d) = HG([HG([HG([...HG([HG([n, b], b-1, c, d), b], b-1, c, d)..., b], b-1, c, d), b], b-1, c, d), b], b-1, c, d) with n nestings
and so on, with any number of entries
This continues in the same way for HG([n, b, c]), HG([n, b, c, d]), and with any number of entries.
HG([n][2]) = HG([n, n, n...n, n, n]) with n n's
HG([n][b]) = HG([HG([HG([...HG([HG([n][b-1])][b-1])...][b-1])][b-1])][b-1]) with n nestings
HG([n][1][2]) = HG([n][HG([n][HG([n][...HG([n][HG([n][n])])...])])]) with n nestings
HG([n][2][2]) = HG([HG([HG([...HG([HG([n][1][2])][1][2])...][1][2])][1][2])][1][2]) with n nestings
HG([n][b][2]) = HG([HG([HG([...HG([HG([n][b-1][2])][b-1][2])...][b-1][2])][b-1][2])][b-1][2]) with n nestings
HG([n][1][c]) = HG([n][HG([n][HG([n][...HG([n][HG([n][n][c-1])][c-1])...][c-1])][c-1])][c-1]) with n nestings
HG([n][b][c]) = HG([HG([HG([...HG([HG([n][b-1][c])][b-1][c])...][b-1][c])][b-1][c])][b-1][c]) with n nestings
HG([n][1][1][2]) = HG([n][n][HG([n][n][HG([n][n][...HG([n][n][HG([n][n][n])])...])])]) with n nestings
HG([n][2][1][2] = HG([HG([HG([...HG([HG([n][1][1][2])][1][1][2])...][1][1][2])][1][1][2])][1][1][2]) with n nestings
HG([n][b][1][2]) = HG([HG([HG([...HG([HG([n][b-1][1][2])][b-1][1][2])...][b-1][1][2])][b-1][1][2])][b-1][1][2]) with n nestings
HG([n][1][2][2]) = HG([HG([HG([...HG([HG([n][n][1][2])][n][1][2])...][n][1][2])][n][1][2])][n][1][2]) with n nestings
HG([n][b][c][d]) = HG([HG([HG([...HG([HG([n][b-1][c][d])][b-1][c][d])...][b-1][c][d])][b-1][c][d])][b-1][c][d]) with n nestings
and so on, with any number of entries
HG([n, 1][2]) = HG([n][n][n]...[n][n][n]) with n [n]'s
HG([n, b][2]) = HG([HG([HG([...HG([HG([n, b-1][2]), b-1][2])..., b-1][2]), b-1][2]), b-1][2]) with n nestings
HG([n, 1][c]) = HG([n, HG([n, HG([n, ...HG([n, HG([n, n][c-1])][c-1])...][c-1])][c-1])][c-1]) with n nestings
HG([n, b][c]) = HG([HG([HG([...HG([HG([n, b-1][c]), b-1][c])..., b-1][c]), b-1][c]), b-1][c]) with n nestings
HG([n, 1, 1][2]) = HG([n, n][HG([n, n][HG([n, n][...HG([n, n][HG([n, n][n])])...])])]) with n nestings
HG([n, 2, 1][2]) = HG([HG([HG([...HG([HG([n, 1, 1][2]), 1, 1][2])..., 1, 1][2]), 1, 1][2]), 1, 1][2]) with n nestings
HG([n, 1, 2][2]) = HG([n, HG([n, HG([n, ...HG([n, HG([n, n, 2][2]), 2][2])..., 2][2]), 2][2]), 2][2]) with n nestings
HG([n, 1, 1][d]) = HG([n, n, HG([n, n, HG([n, n, ...HG([n, n, HG([n, n, n][d-1])][d-1])...][d-1])][d-1])][d-1]) with n nestings
HG([n, b, c][d]) = HG([HG([HG([...HG([HG([n, b-1, c][d]), b-1, c][d])..., b-1, c][d]), b-1, c][d]), b-1, c][d]) with n nestings
and so on, with any number of entries in the first [ ]
HG([n][1, 2]) = HG([n, n, n...n, n, n][n]) with n n's in the first [ ]
HG([n][b, 2]) = HG([HG([HG([...HG([HG([n][b-1, 2])][b-1, 2])...][b-1, 2])][b-1, 2])][b-1, 2]) with n nestings
HG([n][1, c]) = HG([n][HG([n][HG([n][...HG([n][HG([n][n, c-1]), c-1])..., c-1]), c-1]), c-1]) with n nestings
HG([n][b, c]) = HG([HG([HG([...HG([HG([n][b-1, c])][b-1, c])...][b-1, c])][b-1, c])][b-1, c]) with n nestings
HG([n][1, 1, 2]) = HG([n][n, HG([n][n, HG([n][n, ...HG([n][n, HG([n][n, n])])...])])]) with n nestings
HG([n][b, 1, 2]) = HG([HG([HG([...HG([HG([n][b-1, 1, 2])][b-1, 1, 2])...][b-1, 1, 2])][b-1, 1, 2])][b-1, 1, 2]) with n nestings
HG([n][1, 2, 2]) = HG([n][HG([n][HG([n][...HG([n][HG([n][n, 1, 2]), 1, 2])..., 1, 2]), 1, 2]), 1, 2]) with n nestings
HG([n][1, 1, d]) = HG([n][1, HG([n][1, HG([n][1, ...HG([n][1, HG([n][1, n, d-1]), d-1])..., d-1]), d-1]), d-1]) with n nestings
HG([n][b, c, d]) = HG([HG([HG([...HG([HG([n][b-1, c, d])][b-1, c, d])...][b-1, c, d])][b-1, c, d])][b-1, c, d]) with n nestings
and so on, with any number of entries in the second [ ]
HG([n, 1][1, 2]) = HG([n][n, n, n... n, n, n]) with n n's in the second [ ]
This whole process repeats, and repeats, until we get to HG([n, b, c...][d, e, f...]...[h, i, j...][k, l m...]) with any number of [ ] and any number of entries within them.
HG([n[2]]) = HG([n, n, n...][n, n, n...][n, n, n...]...[n, n, n...][n, n, n...][n, n, n...]) with n sets of [ ] with n n's in each
HG([n[b]]) = HG([HG([HG([...HG([HG([n[b-1]])[b-1]])...[b-1]])[b-1]])[b-1]]) with n nestings
HG([n, 1[2]]) = HG([n[HG([n[HG([n[...HG([n[HG([n[n]])]])...]])]])]]) with n nestings
HG([n, b[2]]) = HG([HG([HG([...HG([HG([n, b-1[2]]), b-1[2]])..., b-1[2]]), b-1[2]]), b-1[2]]) with n nestings
HG([n, 1[c]]) = HG([n, HG([n, HG([n, ...HG([n, HG([n, n[c-1]])[c-1]])...[c-1]])[c-1]])[c-1]]) with n nestings
HG([n, b[c]]) = HG([HG([HG([...HG([HG([n, b-1[c]]), b-1[c]])..., b-1[c]]), b-1[c]]), b-1[c]]) with n nestings
This continues in the same way for any number of entries for [n, b...[c...]]
HG([n[1, 2]]) = HG([n, n, n...n, n, n[n]]) with n entries of n
HG([n[1, c]) = HG([n[HG([n[HG([n[...HG([n[HG([n[n]])]])...]])]])]]) with n nestings
HG([n[b, c]]) = HG([HG([HG([...HG([HG([n[b-1, c]])[b-1, c]])...[b-1, c]])[b-1, c]])[b-1, c]]) with n nestings
HG([n[b, c...]]) continues in the same way, with any number of entries
HG([n[1]2]) = HG([n[n, n, n...n, n, n]]) with n entries of n
HG([n[b]2]) = HG([HG([HG([...HG([HG([n[b-1]2])[b-1]2])...[b-1]2])[b-1]2])[b-1]2]) with n nestings
HG([n[1]c]) = HG([n[HG([n[HG([n[...HG([n[HG([n[n]c-1])]c-1])...]c-1])]c-1])]c-1]) with n nestings
HG([n[b]c]) = HG([HG([HG([...HG([HG([n[b-1]c])[b-1]c])...[b-1]c])[b-1]c])[b-1]c]) with n nestings
HG([n, 1[1]2]) = HG([n[n]HG([n[n]HG([n[n]...HG([n[n]HG([n[n]n])])...])])]) with n nestings
This continues in the same way, and with any number of entries
HG([n[1, 1]2]) = HG([n ,n, n...n, n, n[n]n]) with n entries of n
This continues in the same way, and with any number of entries
HG([n[1]1, 2]) = HG([n[n, n, n...n, n, n]n]) with n entries of n
This continues in the same way, and with any number of entries
HG([n|2]) = HG([n[n]n, n, n...n, n, n]) with n entries of n
HG([n|b]) = HG([HG([HG([...HG([HG([n|b-1])|b-1])...|b-1])|b-1])|b-1]) with n nestings
HG([n|1|2]) = HG([n|HG([n|HG([n|...HG([n|HG([n|n])])...])])]) with n nestings
HG([n|b|2]) = HG([HG([HG([...HG([HG([n|b-1|2])|b-1|2])...|b-1|2])|b-1|2])|b-1|2]) with n nestings
HG([n|1|c]) = HG([n|HG([n|HG([n|...HG([n|HG([n|n|c-1])|c-1])...|c-1])|c-1])|c-1]) with n nestings
HG([n|b|c]) = HG([HG([HG([...HG([HG([n|b-1|c])|b-1|c])...|b-1|c])|b-1|c])|b-1|c]) with n nestings
HG([n|1|1|2]) = HG([n|n|HG([n|n|HG([n|n|...HG([n|n|HG([n|n|n])])...])])]) with n nestings
this continues in the same way, with any number of separators
HG([n||2]) = HG([n|n|n...n|n|n]) with n n's
HG([n||b]) = HG([HG([HG([...HG([HG([n||b-1])||b-1])...||b-1])||b-1])||b-1]) with n nestings
Written as HP(n, m), Hexapraginal numbers are Hexagraphs (HG(n)) that contain at least m n's within its digits. Because none of the Hexagraph values have been 100% confirmed, no Hexapyraginal numbers are known.
Examples:
HP(1, 1) is the first HG(n) that contains a single 1 in its digits
HP(12, 2) is the first HG(n) that contains at least 2 12's in its digits
HP(100, 100) is the first HG(n) that contains at least 100 100's in its digits
Written as FP(n, m), Hexapraginal numbers are Hexagraphs (HG(n)) that contain at least m FP(n-1, m)'s within its digits. Because none of the Hexagraph values have been 100% confirmed, no Fixexapraginal numbers are known.
Examples:
FP(1, 1) is the first HG(n) that contains a single 1 in its digits
FP(12, 2) is the first HG(n) that contains at least 2 FP(11, 2)'s in its digits
FP(100, 100) is the first HG(n) that contains at least 100 FP(99, 100)'s in its digits
My third attempt at creating a TREE/SCG/Hydra type sequence that generates enormous numbers, the first being GOSS, then Hexagraphs.
For HMG(n) make a sequence of graphs starting with g(1) (then continues for g(x)). Each graph is (x+2)^n grid of dots, with the power of n meaning n dimensions. The "player" uses (x+1)^n lines to connect dots for each graph. The goal is to make as many graphs as possible without repeating a shape (group of connected lines), without any diagonal lines, and without any shapes that can be transformed to a previously used shape by adding/removing at most ((x+1)^n)+((x^n)-2x-1) lines (basically, at most ((x+1)^n)+((x^n)-2x-1) changes which can include adding or removing a line). If a shape is a mirrored or rotated variant of an existing shape, it does not count as the same shape For HMG(1), g(1) is a line of 3 dots, each connected by a line, so two lines going left to right. g(2) is a line of 4 dots each connected by a line, so 3 line lines from left to right, but this cannot be made since this can be transformed to g(1) in just one change, so HMG(1) = 1. HMG(2) starts with g(1) being a 3 by 3 grid with 4 total lines to be used to make a shape/shapes, say for example a 1x1 square. This shape is now a shape that cannot be made again, or be transformed into. HMG(2) = 2 since any g(3) would include some shape transformable into the first shape. HMG(3) starts with a 3x3x3 cube of dots starting with 8 lines. HMG(3) is currently unknown. The sequence continues for any HMG(n).
HMG(n, a) = HMGa(n)
HMG(n, 1, 2) = HMG(n, HMG(n, ...HMG(n, HMG(n, n))...)) with n nestings
HMG(n, a, 2) = HMG(HMG(...HMG(HMG(n, a-1, 2), a-1, 2)..., a-1, 2), a-1, 2) with n nestings
HMG(n, 1, b) = HMG(n, HMG(n, ...HMG(n, HMG(n, n, b-1), b-1)..., b-1), b-1) with n nestings
HMG(n, a, b) = HMG(HMG(...HMG(HMG(n, a-1, b), a-1, b)..., a-1, b), a-1, b) with n nestings
HMG(n, 1, 1, 2) = HMG(n, n, HMG(n, n, ...HMG(n, n, HMG(n, n, n))...)) with n nestings
HMG(n, a, 1, 2) = HMG(n, a, HMG(n, a,...HMG(n, a, HMG(n, a, n))...)) with n nestings
HMG(n, a, b, 2) = HMG(n, HMG(n, ...HMG(n, HMG(n, n, b-1), b-1)..., b-1), b-1) with n nestings
HMG(n, a, b, c) = HMG(HMG(...HMG(HMG(n, a-1, b, c), a-1, b, c)..., a-1, b, c), a-1, b, c) with n nestings
and so on, with any number of entries.
HMG([n]) = HMG(n, n, n...n, n, n) with n n's
HMG([n, 2]) = HMG([HMG([HMG([...HMG([HMG([n])])...])])]) with n nestings
HMG([n, a]) = HMG([HMG([HMG([...HMG([n, a-1]), a-1...]), a-1]), a-1]) with n nestings
HMG([n, 1, 2]) = HMG([n, HMG([n, ...HMG([n, HMG([n, n])])]...)]) with n nestings
HMG([n, a, 2]) = HMG([HMG([...HMG([HMG([n, a-1, 2]), a-1, 2])..., a-1, 2]), a-1, 2]) with n nestings
HMG([n, 1, b]) = HMG([n, HMG([n, ...HMG([n, HMG([n, n, b-1]), b-1])..., b-1]), b-1]) with n nestings
HMG([n, a, b]) = HMG([HMG([...HMG([HMG([n, a-1, b]), a-1, b])..., a-1, b]), a-1, b]) with n nestings
HMG([n, 1, 1, 2]) = HMG[(n, n, HMG([n, n, ...HMG([n, n, HMG([n, n, n])])...])]) with n nestings
HMG([n, a, 1, 2]) = HMG([n, a, HMG([n, a,...HMG([n, a, HMG([n, a, n])])...)]) with n nestings
HMG([n, a, b, 2]) = HMG([n, HMG([n, ...HMG([n, HMG([n, n, b-1]), b-1])..., b-1]), b-1]) with n nestings
HMG([n, a, b, c]) = HMG([HMG([...HMG([HMG([n, a-1, b, c]), a-1, b, c])..., a-1, b, c]), a-1, b, c]) with n nestings
and so on, with any number of entries.
HMG([n2]) = HMG([n, n, n...n, n, n]) with n n's
HMG([na]) = HMG([HMG([HMG([...HMG([HMG([na-1])a-1])...a-1])a-1])a-1]) with n nestings