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
KEYWORD
nonn
AUTHOR
Natan Arie Consigli, Apr 30 2016
EXTENSIONS
Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020
STATUS
approved