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A267647 a(n) = g_n(4), where g is the weak Goodstein function defined in A266202.

%I A267647 #13 Jan 11 2020 15:57:47
%S A267647 4,8,9,10,11,11,11,11,11,11,11,10,9,8,7,6,5,4,3,2,1,0
%N A267647 a(n) = g_n(4), where g is the weak Goodstein function defined in A266202.
%C A267647 For more info see A266201 - A266202.
%e A267647 g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;
%e A267647 g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;
%e A267647 g_3(4) = b_4(2*4+1)-1 = 2*5 + 1-1 = 10;
%e A267647 g_4(4) = b_5(2*5)-1= 2*6 - 1 = 11;
%e A267647 g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;
%e A267647 g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;
%e A267647 g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;
%e A267647 g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;
%e A267647 g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;
%e A267647 g_10(4) = b_11(11)-1 = 12-1 = 11;
%e A267647 g_11(4) = b_12(11)-1 = 11-1 = 10;
%e A267647 g_12(4) = b_13(10)-1 = 10-1 = 9;
%e A267647 g_13(4) = b_14(9)-1 = 9-1 = 8;
%e A267647 …
%e A267647 g_21(4) = 0;
%t A267647 g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 4], {n, 0, 21}] (* _Michael De Vlieger_, Mar 18 2016 *)
%o A267647 (PARI) a(n) = {if (n == 0, return (4)); wn = 4; for (k=2, n+1, pd = Pol(digits(wn, k)); wn = subst(pd, x, k+1) - 1; ); wn; }
%o A267647 vector(22, n, n--; a(n)) \\ _Michel Marcus_, Apr 03 2016
%Y A267647 Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A266202: g_n(n); A267648: g_5(n); A266203: a(n) = k such that g_k(n)=0;
%Y A267647 A056193: G_n(4).
%K A267647 nonn,full,fini,easy
%O A267647 0,1
%A A267647 _Natan Arie Consigli_, Mar 17 2016