H* function

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Not to be confused with H function.

Definition

H* Function
Growth rate	\(f_{\omega}(x)\)
Based On	-illions (short scale)

The H* function (defined by SuperJedi224 for generating new -illions) is defined as follows:

• H*(m)=H(m)=10^3m+3
• H*(m,0)=m
• H*(m,n+1)=H^H*(m,n)(m)=H(H(H(...H(m)...))), with H*(m,n) Hs.
• H*(a,...,b,c,1)=H*(a,...,b,c)
• H*(a,...,b,c,d)=H*(a,...,b,H*(a,...,b,c,d-1)) (Although the original definition does not clarify it, the rule is probably supposed to be applied only when d > 1).

In other words:

1. If there is precisely one variable, the function value is the variable-th -illon.
2. if there are precisely two variables:
   1. if the second variable is 0,the function value is the first variable.
   2. else:
      1. Decrease the second variable in 1, and call the new function value Z.
      2. The function value is recursion of the first variable through the H* function Z times.
3. else:
   1. if the last entry is 1, remove it.
   2. else:
      1. Decrease the last variable by 1, and call the new function value Z.
      2. Replace the last two variables with one Z.

Extended definition

Extended H* Function
Growth rate	\(f_{\omega^{2} }(x)\)
Based on	-illions, H* function

The extended H* function (defined by SuperJedi224 for generating very large "-illions") is defined as follows:

• H*₁(a)=H(a)=10^3a+3
• H*_n(a,0)=a
• H*₁(a,b)=H(H(H(...H(a)...))), with H*₁(a,b-1) Hs.
• H*_n(a,b)=H*_n-1(a,a,...a,a) with H*_n(a,b-1) "a"s.
• H*_n(a,...,b,c,1)=H*_n(a,...,b,c)
• H*_n(a,...,b,c,d)=H*_n(a,...,b,H*_n(a,...,b,c,d-1))

Examples

• H*(0) = 10³ = 1,000 = thousand
• H*(1) = 10⁶ = 1,000,000 = million
• H*(2) = 10⁹ = 1,000,000,000 = billion
• H*(10) = 10³³ = decillion
• H*(20) = 10⁶³ = vigintillion
• H*(100) = 10³⁰³ = centillion
• H²(0) = H*(H*(0)) = H*(10³) = millillion
• H²(1) = H*(H*(1)) = H*(10⁶) = micrillion
• H²(2) = H*(H*(2)) = H*(10⁹) = nanillion
• H²(10) = H*(H*(10)) = H*(10³³) = mecillion
• H²(20) = H*(H*(20)) = H*(10⁶³) = meicosillion
• H²(100) = H*(H*(100)) = H*(10³⁰³) = mehectillion
• H³(0) = H*(H*(H*(0))) = H*(H*(10³)) = killamillillion
• H³(1) = H*(H*(H*(1))) = H*(H*(10⁶)) = Killamicrillion
• H³(2) = H*(H*(H*(2))) = H*(H*(10⁹)) = Killananillion
• H³(10) = H*(H*(H*(10))) = H*(H*(10³³))
• H³(20) = H*(H*(H*(20))) = H*(H*(10⁶³))
• H³(100) = H*(H*(H*(100))) = H*(H*(10³⁰³))
• H*(10,1) = H^H*(10,0)(10) = H¹⁰(10) = H*(H*(H*(H*(H*(H*(H*(H*(H*(H*(10)))))))))) = hyperillion
• H*(20,1) = H^H*(20,0)(20) = H²⁰(20) = H*(H*(...H*(20)...)) (20 H's) = icosi-hyperillion
• H*(100,1) = H^H*(100,0)(100) = H¹⁰⁰(100) = H*(H*(...H*(100)...)) (100 H's) = cent-hyperillion
• H*(10,2) = H^H*(10,1)(10) = H^H10(10)(10) = H*(H*(...H*(10)...)) (hyperillion H's) = grand hyperillion
• H*(20,2) = H^H*(20,1)(20) = H^H20(20)(20) = H*(H*(...H*(20)...)) (icosi-hyperillion H's) = grand-icosi-hyperillion
• H*(10,3) = H^H*(10,2)(10) = H^HH10(10)(10)(10) = H*(H*(...H*(10)...)) (grand hyperillion H's) = bigrand hyperillion
• H*(20,3) = H^H*(20,2)(10) = H^HH20(20)(20)(20) = H*(H*(...H*(20)...)) (grand-icosi-hyperillion H's) = bigrand-icosi-hyperillion
• H*(10,4) = H^H*(10,3)(10) = H^{HHH10(10)(10)(10)}(10) = H*(H*(...H*(10)...)) (bigrand hyperillion H's) = trigrand hyperillion
• H*(10,10) = H^H*(10,9)(10) = H^{HHHHHHHHH10(10)(10)(10)(10)(10)(10)(10)(10)(10)}(10) = hyperduillion
• H*(10,10,2) = H*(10,H*(10,10)) = grand hyperduillion
• H*(10,10,3) = H*(10,H*(10,H*(10,10))) = bigrand hyperduillion
• H*(10,10,10) = H*(10,H*(10,...H*(10,10)...)) (10 H's) = hypertrillion
• H*(10,10,10,10) = H*(10,10,H*(10,10,...H*(10,10,10)...)) (10 H's) = hyperterillion
• H*(10,10,10,10,10) = H*(10,10,10,H*(10,10,10,...H*(10,10,10,10)...)) (10 H's) = hyperpetillion
• H*₂(10,1) = H*(10,10,10,10,10,10,10,10,10,10) = H*(10,10,10,10,10,10,10,10,H*(10,10,10,10,10,10,10,10,...H*(10,10,10,10,10,10,10,10,10)...)) (10 H's) = hyperhypillion
• H*₂(10,2) = H*(10,10,...,10,10) (hyperhypillion amount of tens) = grand hyperhypillion
• H*₂(10,3) = H*(10,10,...,10,10) (grand hyperhypillion amount of tens) = bigrand hyperhypillion
• H*₂(10,10) = great hyperhypillion
• H*₂(100,100) = great cent-hyperhypillion