The hyperlicious function is defined as
- \(h_x(a,b) = a\uparrow^{x-2} b \),
- \(h_x(a,b,c,\ldots,m,n,1) = h_x(a,b,c,\ldots,m,n)\), and
- \(h_x(a,b,c,\ldots,m,n) = h_{h_x(a,b,c,\ldots,m - 1)}(a,b,c,\ldots,m)\).[1][dead link]
Up to \(h_x(a,b)\)[]
- \(h_0(a,b) = a+1\) (successor function)
- \(h_1(a,b) = a+b\) (addition)
- \(h_2(a,b) = ab\) (multiplication)
- \(h_3(a,b) = a^b\) (exponentiation)
- \(h_4(a,b) = a \uparrow\uparrow b\) (tetration)
- \(h_5(a,b) = a \uparrow\uparrow\uparrow b\) (pentation)
- \(h_6(a,b) = a \uparrow\uparrow\uparrow\uparrow b\) (hexation)
- \(h_7(a,b) = a \uparrow\uparrow\uparrow\uparrow\uparrow b\) (heptation)
- \(h_8(a,b) = a \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow b\) (octation)
- \(h_9(a,b) = a \uparrow^7 b\) (enneation)
- \(h_{10}(a,b) = a \uparrow^8 b\) (decation)
- \(h_{20}(a,b) = a \uparrow^{18} b\) (vigintation)
- \(h_{100}(a,b) = a \uparrow^{98} b\) (centation)
Examples[]
- \(h_3(2,6) = 2 \uparrow 6\)
- \(h_3(10,100) = 10 \uparrow 100\) (googol)
- \(h_4(10,100) = 10 \uparrow\uparrow 100\) (giggol)
- \(h_5(10,100) = 10 \uparrow\uparrow\uparrow 100\) (gaggol)
- \(h_4(2,6,2) = h_{h_4(2,5)}(2,6) = h_{2\uparrow^6 5}(2,6) = 2 \uparrow^{2 \uparrow^{6} 5-2} 6\)
Growth rate[]
\(h(x) = h_x(x,x...x,x)\) with x x's in the array is approximately \(f_{\omega+1}(x)\) in the fast growing hierarchy, which is closely related to the expansion function ({x,x,1,2}). The two-entry hyperlicious function (\(h_x(a,b)\)) eventually dominates any hyper-operator, such as tetration, pentation, or even beyond centation.