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g_n(8) where g is the weak Goodstein function defined in A266202.
3

%I #10 Sep 29 2023 08:01:32

%S 8,26,41,60,83,109,139,173,211,253,299,348,401,458,519,584,653,726,

%T 803,884,969,1058,1151,1222,1295,1370,1447,1526,1607,1690,1775,1862,

%U 1951,2042,2135,2230,2327,2426,2527,2630,2735,2842,2951,3062,3175,3290,3407,3525,3645,3767,3891,4017,4145,4275,4407,4541

%N g_n(8) where g is the weak Goodstein function defined in A266202.

%C For more info see A266201-A266202.

%e g_1(8) = b_2(8)-1 = b_2(2^3)-1 = 3^3-1 = 26;

%e g_2(8) = b_3(2*3^2+2*3+2)-1 = 2*4^2+2*4+2-1 = 41;

%e g_3(8) = b_4(2*4^2+2*4+1)-1 = 2*5^2+2*5+1-1 = 60;

%e g_4(8) = b_5(2*5^2+2*5)-1 = 2*6^2+2*6-1 = 83;

%e g_5(8) = b_6(2*6^2+6+5)-1 = 2*7^2+7+5-1 = 109;

%e g_6(8) = b_7(2*7^2+7+4)-1 = 2*8^2+8+4-1 = 139;

%e g_7(8) = b_8(2*8^2+8+3)-1 = 2*9^2+9+3-1 = 173;

%e g_8(8) = b_9(2*9^2+9+2)-1 = 2*10^2+10+2-1 = 211;

%e g_9(8) = b_10(2*10^2+10+1)-1 = 2*11^2+11+1-1 = 253;

%e g_10(8) = b_11(2*11^2+11)-1 = 2*12^2+12-1 = 299.

%t 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, 8], {n, 0, 55}]

%Y Essentially the same as A056193.

%Y Cf. G_n(8): A271555.

%Y Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

%K nonn,fini

%O 0,1

%A _Natan Arie Consigli_, May 22 2016