login
g_n(7) where g is the weak Goodstein function defined in A266202.
4

%I #7 Jan 11 2020 15:57:47

%S 7,12,19,27,37,49,63,69,75,81,87,93,99,105,111,116,121,126,131,136,

%T 141,146,151,156,161,166,171,176,181,186,191,195,199,203,207,211,215,

%U 219,223,227,231,235,239,243,247,251,255,259,263,267,271,275,279,283,287,291,295,299,303,307,311,315,319,322,325

%N g_n(7) where g is the weak Goodstein function defined in A266202.

%C For more info see A266201-A266202.

%e g_1(7)= b_2(7)-1 = b_2(2^2+2+1)-1 = 3^2+3+1-1 = 12;

%e g_2(7) = b_3(3^2+3)-1 = 4^2+4-1 = 19;

%e g_3(7) = b_4(4^2+3)-1 = 5^2+3-1 = 27;

%e g_4(7) = b_5(5^2+2)-1 = 6^2+2-1 = 37;

%e g_5(7) = b_6(6^2+1)-1 = 7^2+1-1 = 49;

%e g_6(7) = b_7(7^2)-1 = 8^2-1 = 63;

%e g_7(7) = b_8(7*8+7)-1 = 7*9+7-1 = 69;

%e ...

%e g_2045(7) = 0.

%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, 7], {n, 0, 64}]

%Y Cf. A271554: G_n(7).

%Y Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); 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 21 2016