A271985 - OEIS

%I #22 Sep 25 2020 11:15:49

%S 0,253,4382,885775,150051213,570623341475,855935016215,1426559238830,

%T 1997331745490,3138428376974,3138428381103,3138429262496,3138578427934

%N G_9(n), where G is the Goodstein function defined in A266201.

%C a(17) = 2.066...*10^4574. - _Pontus von Brömssen_, Sep 25 2020

%H Pontus von Brömssen, <a href="/A271985/b271985.txt">Table of n, a(n) for n = 3..16</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's theorem</a>

%e Compute G_9(10):

%e G_1(10)= B_2(10)-1 = B_2(2^(2+1)+2)-1 = 3^(3+1)+3-1 = 83;

%e G_2(10) = B_3(3^(3+1)+2)-1 = 4^(4+1)+2-1 = 1025;

%e G_3(10) = B_4(4^(4+1)+1)-1 = 5^(5+1)+1-1 = 15625;

%e G_4(10) = B_5(5*5^(5+1))-1 = 6^(6+1)-1= 279935;

%e G_5(10) = B_6(5*6^6+5*6^5+5*6^4+5*6^3+5*6^2+5*6+5)-1 = 5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+5-1 = 4215754;

%e G_6(10) = B_7(5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+4)-1 = 5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+4-1 = 84073323;

%e G_7(10) = B_8(5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+3)-1 = 5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+3-1 = 1937434592;

%e G_8(10) = B_9(5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+2)-1 = 5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+2-1 = 50000555551;

%e G_9(10) = B_10(5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+1)-1 = 5*11^11+5*11^5+5*11^4+5*11^3+5*11^2+5*11+1-1 = 1426559238830.

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o def bump(n,b):

%o   s=digits(n,b)[1:]

%o   l=len(s)

%o   return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])

%o def A271985(n):

%o   if n==3: return 0

%o   for i in range(2,11):

%o     n=bump(n,i)-1

%o   return n # _Pontus von Brömssen_, Sep 25 2020

%Y 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); A271979: G_8(n); this sequence: G_9(n); A271986: G_10(n); A266201: G_n(n).

%K nonn

%O 3,2

%A _Natan Arie Consigli_, Apr 30 2016

%E Incorrect program and terms removed by _Pontus von Brömssen_, Sep 25 2020