cp's OEIS Frontend

A268689 Let f(n) = maximal value of the weak Goodstein function g_k(n) for k >= 0; then a(n) = minimal value of k such that g_k(n) = f(n).

%I A268689 #16 Jan 11 2020 15:57:47
%S A268689 0,0,0,0,4,14,94,510
%N A268689 Let f(n) = maximal value of the weak Goodstein function g_k(n) for k >= 0; then a(n) = minimal value of k such that g_k(n) = f(n).
%C A268689 g_k(n) is the weak Goodstein function defined in A266202.
%e A268689 g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;
%e A268689 g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;
%e A268689 g_3(4) = b_4(2*4 + 1 ) -1 = 2*5 + 1-1 = 10;
%e A268689 g_4(4) = b_5(2*5) -1= 2*6 - 1 = 11;
%e A268689 g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;
%e A268689 g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;
%e A268689 g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;
%e A268689 g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;
%e A268689 g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;
%e A268689 g_10(4) = b_11(11)-1 = 12-1 = 11;
%e A268689 g_11(4) = b_12(11)-1 = 11-1 = 10;
%e A268689 g_12(4) = b_13(10)-1 = 10-1 = 9;
%e A268689 g_13(4) = b_14(9)-1 = 9-1 = 8;
%e A268689 …
%e A268689 g_21(4) = 0;
%e A268689 So a(4) = 4.
%o A268689 (PARI) g(n, k) = {if (n == 0, return (k)); wn = k; for (k=2, n+1, pd = Pol(digits(wn, k)); wn = subst(pd, x, k+1) - 1; ); wn; }
%o A268689 a(n) = {vg = []; ok = 1; ns = 0; while(ok, newg = g(ns, n); vg = concat(vg, newg); if (newg <= 0, ok = 0); ns++;); vmax = vecmax(vg); k = 1; while (vg[k] != vmax, k++); k--;} \\ _Michel Marcus_, Apr 03 2016
%Y A268689 Cf. A266203, A268687, A268688.
%K A268689 nonn,hard
%O A268689 0,5
%A A268689 _Natan Arie Consigli_, Apr 03 2016
%E A268689 a(7) from _Michel Marcus_, Apr 03 2016