A372237 - OEIS

%I #12 Apr 25 2024 12:28:23

%S 4,15,26,49,96,191,318,573,1084,2107,4154,8249,16440,32823,65590,

%T 131125,262196,524339,1048626,2097201,4194352,8388655,16777262,

%U 33554477,67108908,134217771,268435498,536870953,1073741864,2147483687,4294967334,8589934629,17179869220

%N a(0) = 4; to obtain a(k), write out the base-(2^k) expansion of a(k-1), bump to base 2^(k+1), then subtract 1.

%C Applying to the proof of the usual Goodstein's theorem to the ordinal number omega^omega shows that: for no matter what initial value and no matter what increasing sequence of bases b(0), b(1), ... with b(0) >= 2, the (weak) Goodstein sequence eventually terminates with 0. Here b(k) = 2^(k+1).

%C Sequence terminates at a(2^(2^70+70) + 2^70 + 68) = 0.

%H Jianing Song, <a href="/A372237/b372237.txt">Table of n, a(n) for n = 0..1000</a>

%H Googology Wiki, <a href="http://googology.wikia.com/wiki/Goodstein_sequence">Goodstein sequence</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>

%F a(k) = 2^(k+2) + 68 - k for 5 <= k <= 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 2 and 68 - k.

%F a(k) = 2^(k+1) + 2^70 + 68 - k for 69 <= 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 1 and 2^70 + 68 - k.

%F a(k) = 2^(2^70+70) + 2^70 + 68 - k for 2^70 + 69 <= k <= 2^(2^70+70) + 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of a single digit  2^(2^70+70) + 2^70 + 68 - k.

%e a(0) = 100_2 = 4;

%e a(1) = 100_4 - 1 = 15 = 33_4;

%e a(2) = 33_8 - 1 = 26 = 32_8;

%e a(3) = 32_16 - 1 = 49 = 31_16;

%e a(4) = 31_32 - 1 = 96 = 30_32;

%e a(5) = 30_64 - 1 = 191 = (2,63)_64.

%o (PARI) A372237_first_N_terms(N) = my(v=vector(N+1)); v[1] = 4; for(i=1, N, v[i+1] = fromdigits(digits(v[i],2^i),2^(i+1))-1); v

%Y Cf. A056193, A266202.

%K nonn,easy,fini

%O 0,1

%A _Jianing Song_, Apr 23 2024