A266201 Goodstein numbers: a(n) = G_n(n), where G is the Goodstein function.

0, 0, 1, 2, 83, 1197, 187243, 37665879, 20000000211, 855935016215, 44580503598539, 2120126221988686, 155568095557812625, 6568408355712901455, 295147905179358418247, 14063084452070776884879

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..15.

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic, Vol. 9, No. 2 (1944), pp. 33-41; alternative link.

Eric Weisstein's World of Mathematics, Goodstein sequences.

Example

Compute a(5) = G_5(5):
G_0(5) = 5;
G_1(5) = B_2(G_0(5))-1 = B_2(2^2+1)-1 = (3^3+1)-1 = 27 = 3^3;
G_2(5) = B_3(G_1(5))-1 = B_3(3^3)-1 = 4^4-1 = 255 = 3*4^3+3*4^2+3*4+3;
G_3(5) = B_4(G_2(5))-1 = B_4(3*4^3+3*4^2+3*4+3)-1 = 467;
G_4(5) = B_5(G_3(5))-1 = B_5(3*5^3+3*5^2+3*5+2)-1 = 775;
G_5(5) = B_6(G_4(5))-1 = B_6(3*6^3+3*6^2+3*6+1)-1 = 1197.

Prog

(PARI) (B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n<b+i, #n-i, B(#n-i, b)))); A266201(n)=for(k=1, n, n=B(n, k+1)-1); n \\ M. F. Hasler, Feb 12 2017

Crossrefs

Cf. Goodstein sequences: A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); A215409: G_n(3); A056193: G_n(4); A266204: G_n(5); A266205: G_n(6); A222117: G_n(15); A059933: G_n(16); A211378: G_n(19).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A266202: g_n(n); A266203: a(n) = k such that g_k(n)=0;

Bumping Sequences: A222112: B_2(n);

Other sequences: A222113.

Sequence in context: A171399 A037069 A065591 * A225807 A232770 A317724

Adjacent sequences: A266198 A266199 A266200 * A266202 A266203 A266204

Keyword

nonn

Author

Natan Arie Consigli, Jan 22 2016

Extensions

Edited by M. F. Hasler, Feb 12 2017

Incorrect a(16) deleted (the correct value is ~ 2.77*10^861) by M. F. Hasler, Feb 19 2017

Status

approved