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The maximum values of k where g_k(n) is the maximal value.

g_k(n) is the weak Goodstein function defined in A266202.

Next term: 3*2^402653210-1.

To write an integer n in base-k hereditary representation, write n in ordinary base-k representation, and then do the same recursively for all exponents which are greater than k.

For example, the hereditary representation of 132132 in base-2 is:

132132 = 2^17 + 2^10 + 2^5 + 2^2

= 2^(2^4 + 1) + 2^(2^3 + 2) + 2^(2^2 + 1) + 2^2

= 2^(2^(2^2) + 1) + 2^(2^(2+1) + 2) + 2^(2^2 + 1) + 2^2.

Define B_k(n) to be the function that substitutes k+1 for all the bases of the base-k hereditary representation of n.

E.g., B_2(101) = B_2(2^(2^2 + 2) + 2^(2^2 + 1) + 2^2 + 1) = 3^(3^3 + 3) + 3^(3^3 + 1) + 3^3 + 1 = 228767924549638.

(Sometimes B_k(n) is referred to as n "bumped" from base k.)

The Goodstein function is defined as: G_k(n) = B_{k+1}(G_{k-1}(n)) - 1 with G_0(n) = n, i.e., iteration of bumping the number to the next larger base and subtracting one; see example section for instances.

Goodstein's theorem says that for any nonnegative n, the sequence G_k(n) eventually stabilizes and then decreases by 1 in each step until it reaches 0. (The subsequent values of G_k(n) < 0 are not part of the sequence.)

Named after the English mathematician Reuben Louis Goodstein (1912-1985). - Amiram Eldar, Jun 19 2021

A nonnegative n in ordinary (depth-1) base-k representation is n rewritten as a linear combination k powers n = n_1*b^m_1 + ... + n_k*b^m_k where 0 < n_i < b and m_1 > ... > m_k >= 0.

For instance, the ordinary representation of 34 in base 3 is 3^3 + 2*3 + 1.

Let b_k(n) be the function that substitutes the bases of the base-k representation of n with the base k+1. E.g., b_3(34) = b_3(3^3 + 2*3 + 1) = 4^3 + 2*4 + 1 = 73.

Define the weak Goodstein function as: g_k(n) = b_(k+1)(g_(k-1)(n))-1, g_0(n) = n.

See example for instances.

Let n be a fixed nonnegative integer: Goodstein's theorem shows that the sequence g_k(n) eventually stabilizes and then decreases by 1 at each step until it reaches 0. Thereafter, all the values of g_k(n) < 0 are not part of the sequence.

By Goodstein's theorem we conclude that g_k(n) is a finite sequence.

The sequence eventually goes to zero, as can be seen by noting that multiples of the highest exponent (3 in this case) only go down; in fact the 8th term, a(8) = 7*8^2 + 7*8 + 7 = 511; after which the multiple of the square term will only go down, etc.

This sequence, for 11, grows beyond the quintillions of digits before going to zero.

From Zhuorui He, Aug 07 2025: (Start)

For more info see A266201-A266202.

This sequence has A266203(11)+1 terms and a(A266203(11))=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(11)-1)/2)=(A266203(11)+1)/2. 10^^8 < A266203(11) < 10^^9.

More precisely, 10^(10^(10^(10^(10^(10^(10^619.29937)))))) < A266203(11) < 10^(10^(10^(10^(10^(10^(10^619.299371)))))). (End)

For more info see A266201 - A266202.

For more info see A266201-A266202.

This sequence has A266203(5)+1=62 terms and a(A266203(5))=a(61)=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(5)-1)/2)=a(30)=(A266203(5)+1)/2=31. - Zhuorui He, Aug 08 2025

a(8)=3*2^(3*2^27+27)-1 which is more than 10^(10^8) and equal to the final base of the Goodstein sequence starting with g(2)=4; indeed, apart from the initial term, the sequence starting with b(2)=8 is identical to the Goodstein sequence starting with g(2)=4. The initial terms of a(n) [2, 3, 5 and 7] are equal to the initial terms of the equivalent final bases of Goodstein sequences starting at the same points. a(9)=2^(2^(2^70+70)+2^70+70)-1 which is more than 10^(10^(10^20)).

It appears that if n is even then a(n) is one less than three times a power of two, while if n is odd then a(n) is one less than a power of two.

Comment from John Tromp, Dec 02 2004: The sequence 2,3,5,7,3*2^402653211 - 1, ... gives the final base of the Goodstein sequence starting with n. This is an example of a very rapidly growing function that is total (i.e. defined on any input), although this fact is not provable in first-order Peano Arithmetic. See the links for definitions. This grows even faster than the Friedman sequence described in the Comments to A014221.

In fact there are two related sequences: (i) The Goodstein function l(n) = number of steps for the Goodstein sequence to reach 0 when started with initial term n >= 0: 0, 1, 3, 5, 3*2^402653211 - 3, ...; and (ii) the same sequence + 2: 2, 3, 5, 7, 3*2^402653211 - 1, ..., which is the final base reached. Both grow too rapidly to have their own entries in the database.

Related to the hereditary base sequences - see cross-reference lines.

This sequence gives the final base of the weak Goodstein sequence starting with n; compare A266203, the length of the weak Goodstein sequence. a(n) = A266203(n) + 2.

For more info see A266201-A266202.

For more info see A266201-A266202.

For more info see A266201-A266202.