A271979 G_8(n), where G is the Goodstein function defined in A266201.

0, 211, 3325, 555551, 77777775, 20000000211, 30000003325, 50000555551, 70077777775, 100000000211, 100000003325, 100000555551, 100077777775

Offset

3,2

Comments

At least half of the digits of every term (except a(14)) are the same.

Let n > 0:

a(4n) mod 100 = 211;

a(4n+1) mod 1000 = 3325;

a(4n+2) mod 1000000 = 555551;

a(4n+3) mod 100000000 = 77777775;

Proof for a(4n):

If x is divisible by 4 its hereditary representation in base 2 has all summands divisible by 4 and it cannot have the summands 1 and 2.

If we calculate G_1(x) we would end with:

G_1(x) = B_2(x)-1.

Clearly, B_2(x) = 3^a + 3^b + ... is divisible by 3^3 = 27 and that would mean that the representation of B_2(x)-1 would be B_2(x)-1 = X_3 + 2*3^2+2*3+2.

From now on, let X_n be a sum of powers of n (greater than the right term).

We finish proving the statement by calculating G_8(x):

G_2(x) = B_3(X_3 +2*3^2+2*3+2)-1 = X_4 + 2*4^2+2*4+2-1;

G_3(x) = B_4(X_4 +2*4^2+2*4-1)-1 = X_5 + 2*5^2+2*5+1-1;

G_4(x) = B_5(X_5 +2*5^2+2*5)-1 = X_6 + 2*6^2+2*6-1;

G_5(x) = B_6(X_6 +2*6^2+6+5)-1 = X_7 + 2*7^2+7+5-1;

G_6(x) = B_7(X_7 +2*7^2+7+4)-1 = X_8 + 2*8^2+8+4-1;

G_7(x) = B_8(X_8 +2*8^2+8+3)-1 = X_9 + 2*9^2+9+3-1;

G_8(x) = B_9(X_9 +2*9^2+9+2)-1 = X_10 + 2*10^2+10+2-1 = X_10 + 211;

So finally G_8(x) mod 100 = 211.

The other cases can be proved using the same reasoning.

a(17) = 3.3330...*10^3333, a(18) = 5.555550...*10^555555. - Pontus von Brömssen, Sep 25 2020

Links

Pontus von Brömssen, Table of n, a(n) for n = 3..16

Wikipedia, Goodstein's theorem

Example

Calculate G_8(5):
G_1(5) = B_2(5)-1 = B_2(2^2+1)-1 = 27;
G_2(5) = B_3(3^3)-1 = 4^4-1 = 255;
G_3(5) = B_4(3*4^3 + 3*4^2 + 3*4 + 3)-1 = 3*5^3 + 3*5^2 + 3*5 + 3-1 = 467;
G_4(5) = B_5(3*5^3 + 3*5^2 + 3*5 + 2)-1 = 3*6^3 + 3*6^2 + 3*6 + 2-1 = 775;
G_5(5) = B_6(3*6^3 + 3*6^2 + 3*6 + 1)-1 = 3*7^3 + 3*7^2 + 3*7 + 1-1 = 1197;
G_6(5) = B_7(3*7^3 + 3*7^2 + 3*7)-1 = 3*8^3 + 3*8^2 + 3*8-1 = 1751;
G_7(5) = B_8(3*8^3 + 3*8^2 + 2*8 + 7)-1 = 3*9^3 + 3*9^2 + 2*9 + 7-1 = 2454;
G_8(5) = B_9(3*9^3 + 3*9^2 + 2*9 + 6)-1 = 3*10^3 + 3*10^2 + 2*10 + 6-1 = 3325.

Prog

(Python)
from sympy.ntheory.factor_ import digits
def bump(n, b):
  s=digits(n, b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1, b) for i in range(l) if s[i])
def A271979(n):
  if n==3: return 0
  for i in range(2, 10):
    n=bump(n, i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); A271977: G_6(n); A271978: G_7(n); this sequence: G_8(n); A271985: G_9(n); A271986: G_10(n); A266201: G_n(n).

Sequence in context: A291075 A330426 A071367 * A185719 A224101 A013529

Adjacent sequences: A271976 A271977 A271978 * A271980 A271981 A271982

Keyword

nonn

Author

Natan Arie Consigli, Apr 30 2016

Extensions

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

Status

approved