A271557 - OEIS

%I #24 Jan 11 2020 15:57:47

%S 10,83,1025,15625,279935,4215754,84073323,1937434592,50000555551,

%T 1426559238830,44580503598539,1514375534972427,55560034130686045,

%U 2189469451908364943,92233720368553350471,4136201309431691363859,196732040376482697880697,9892098278301567958688175

%N a(n) = G_n(10), where G is the Goodstein function defined in A266201.

%H Nicholas Matteo, <a href="/A271557/b271557.txt">Table of n, a(n) for n = 0..384</a>

%H R. L. Goodstein, <a href="http://www.jstor.org/stable/2268019">On the Restricted Ordinal Theorem</a>, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

%e G_1(10) = B_2(10)-1 = B_2(2^(2+1)+2)-1 = 3^(3+1)+3-1 = 83;

%e G_2(10) = B_3(3^(3+1)+2)-1 = 4^(4+1)+2-1 = 1025;

%e G_3(10) = B_4(4^(4+1)+1)-1 = 5^(5+1)+1-1 = 15625;

%e G_4(10) = B_5(5*5^(5+1))-1 = 6^(6+1)-1= 279935;

%e G_5(10) = B_6(5*6^6+5*6^5+5*6^4+5*6^3+5*6^2+5*6+5)-1 = 5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+5-1 = 4215754;

%e G_6(10) = B_7(5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+4)-1 = 5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+4-1 = 84073323;

%e G_7(10) = B_8(5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+3)-1 = 5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+3-1 = 1937434592;

%e G_8(10) = B_9(5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+2)-1 = 5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+2-1 = 50000555551.

%o (PARI) lista(nn) = {print1(a = 10, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

%Y Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A266201: G_n(n).

%K nonn,fini

%O 0,1

%A _Natan Arie Consigli_, Apr 11 2016