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Number of steps k to make g_k(n) converge to zero.
14

%I #19 Jan 11 2020 15:57:47

%S 0,1,3,5,21,61,381,2045

%N Number of steps k to make g_k(n) converge to zero.

%C Next term is 3*2^402653211 - 3;

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

%C For a complete table click the link below, and see table of upper bounds on weak Goodstein sequence.

%H Googology Wiki, <a href="http://googology.wikia.com/wiki/Goodstein_sequence">Weak Goodstein Table</a>

%F a(n) = k such that g_k(n)=0.

%e Find a(4):

%e g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;

%e g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;

%e g_3(4) = b_4(2*4 + 1 ) -1 = 2*5 + 1-1 = 10;

%e g_4(4) = b_5(2*5) -1= 2*6 - 1 = 11;

%e g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;

%e g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;

%e g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;

%e g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;

%e g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;

%e g_10(4) = b_11(11)-1 = 12-1 = 11;

%e g_11(4) = b_12(11)-1 = 11-1 = 10;

%e g_12(4) = b_13(10)-1 = 10-1 = 9;

%e g_13(4) = b_14(9)-1 = 9-1 = 8;

%e …

%e g_21(4) = 0 so a(4)=21.

%Y Cf. A266202.

%K nonn,hard

%O 0,3

%A _Natan Arie Consigli_, Jan 22 2016