editing

approved

Harvey P. Dale, <a href="/A137411/b137411.txt">Table of n, a(n) for n = 2..1000</a>

nxt[{n_, a_}]:={n+1, FromDigits[IntegerDigits[a, n+1], n+2]-1}; Transpose[ NestList[ nxt, {1, 11}, 50]][[2]] (* Harvey P. Dale, Feb 09 2015 *)

approved

editing

To obtain a(n + 1), write a(n) in base n, increase the base to n + 1, and subtract 1.

nonn,new

nonn

Weak Goodstein sequence starting at 11.

11, 30, 67, 127, 217, 343, 511, 636, 775, 928, 1095, 1276, 1471, 1680, 1903, 2139, 2389, 2653, 2931, 3223, 3529, 3849, 4183, 4531, 4893, 5269, 5659, 6063, 6481, 6913, 7359, 7818, 8291, 8778, 9279, 9794, 10323, 10866, 11423, 11994, 12579, 13178

2,1

The sequence eventually goes to zero, as can be seen by noting that multiples of the highest exponent (3 in this case) only go down; in fact the 8th term, a(8) = 7*8^2 + 7*8 + 7 = 511; after which the multiple of the square term will only go down, etc.

This sequence, for 11, grows beyond the quintillions of digits before going to zero.

K. Hrbacek & T. Jech, Introduction to Set Theory, Taylor & Francis Group, 1999, pp. 125-127.

To obtain a(n + 1), write a(n) in base n, increase the base to n + 1, and subtract 1.

a(2) = 11 = 2^3 + 2^1 + 2^0

a(3) = 3^3 + 3^1 + 3^0 - 1 = 30

a(4) = 4^3 + 4^1 - 1 = 4^3 + 3*4^0 = 67

Cf. A056004 (strong Goodstein sequences), A059933 (strong Goodstein sequence for 16.).

nonn

Nicholas Matteo (kundor(AT)kundor.org), Apr 15 2008

approved