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A266203 Number of steps k to make g_k(n) converge to zero.

%I A266203 #22 Aug 08 2025 10:30:03
%S A266203 0,1,3,5,21,61,381,2045
%N A266203 Number of steps k to make g_k(n) converge to zero.
%C A266203 Next term is 3*2^402653211 - 3;
%C A266203 g_k(n) is the weak Goodstein function defined in A266202.
%C A266203 For a complete table click the link below, and see table of upper bounds on weak Goodstein sequence.
%H A266203 Googology Wiki, <a href="http://googology.wikia.com/wiki/Goodstein_sequence">Weak Goodstein Table</a>
%F A266203 a(n) = k such that g_k(n)=0.
%F A266203 a(n) = A056041(n)-2. - _Pontus von Brömssen_, Aug 08 2025
%e A266203 Find a(4):
%e A266203 g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;
%e A266203 g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;
%e A266203 g_3(4) = b_4(2*4 + 1 ) -1 = 2*5 + 1-1 = 10;
%e A266203 g_4(4) = b_5(2*5) -1 = 2*6 - 1 = 11;
%e A266203 g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;
%e A266203 g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;
%e A266203 g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;
%e A266203 g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;
%e A266203 g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;
%e A266203 g_10(4) = b_11(11)-1 = 12-1 = 11;
%e A266203 g_11(4) = b_12(11)-1 = 11-1 = 10;
%e A266203 g_12(4) = b_13(10)-1 = 10-1 = 9;
%e A266203 g_13(4) = b_14(9)-1 = 9-1 = 8;
%e A266203 ...
%e A266203 g_21(4) = 0 so a(4)=21.
%Y A266203 Cf. A056041, A266202.
%K A266203 nonn,hard
%O A266203 0,3
%A A266203 _Natan Arie Consigli_, Jan 22 2016