A271555 a(n) = G_n(8), where G is the Goodstein function defined in A266201.

8, 80, 553, 6310, 93395, 1647195, 33554571, 774841151, 20000000211, 570623341475, 17832200896811, 605750213184854, 22224013651116433, 875787780761719208, 36893488147419103751, 1654480523772673528938, 78692816150593075151501, 3956839311320627178248684

Offset: 0

Author

Natan Arie Consigli, Apr 10 2016

Keywords

• nonn
• fini

Examples

			G_1(8) = B_2(8)-1 = B_2(2^(2+1))-1 = 3^(3+1)-1 = 80;
G_2(8) = B_3(2*3^3+2*3^2+2*3+2)-1 = 2*4^4+2*4^2+2*4+2-1 = 553;
G_3(8) = B_4(2*4^4+2*4^2+2*4+1)-1 = 2*5^5+2*5^2+2*5+1-1 = 6310;
G_4(8) = B_5(2*5^5+2*5^2+2*5)-1 = 2*6^6+2*6^2+2*6-1 = 93395;
G_5(8) = B_6(2*6^6+2*6^2+6+5)-1 = 2*7^7+2*7^2+7+5-1 = 1647195;
G_6(8) = B_7(2*7^7+2*7^2+7+4)-1 = 2*8^8+2*8^2+8+4-1 = 33554571;
G_7(8) = B_8(2*8^8+2*8^2+8+3)-1 = 2*9^9+2*9^2+9+3-1 = 774841151.
		

Links

• Nicholas Matteo, Table of n, a(n) for n = 0..384
• R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
• Wikipedia, Goodstein sequence

Crossrefs

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), A266201: G_n(n).

Programs

PARI

lista(nn) = {print1(a = 8, ", "); 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, ", "); ); }

Extensions

a(3) corrected by Nicholas Matteo, Aug 15 2019