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A372237
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.
1
4, 15, 26, 49, 96, 191, 318, 573, 1084, 2107, 4154, 8249, 16440, 32823, 65590, 131125, 262196, 524339, 1048626, 2097201, 4194352, 8388655, 16777262, 33554477, 67108908, 134217771, 268435498, 536870953, 1073741864, 2147483687, 4294967334, 8589934629, 17179869220
OFFSET
0,1
COMMENTS
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).
Sequence terminates at a(2^(2^70+70) + 2^70 + 68) = 0.
LINKS
Googology Wiki, Goodstein sequence.
FORMULA
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.
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.
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.
EXAMPLE
a(0) = 100_2 = 4;
a(1) = 100_4 - 1 = 15 = 33_4;
a(2) = 33_8 - 1 = 26 = 32_8;
a(3) = 32_16 - 1 = 49 = 31_16;
a(4) = 31_32 - 1 = 96 = 30_32;
a(5) = 30_64 - 1 = 191 = (2,63)_64.
PROG
(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
CROSSREFS
Sequence in context: A061873 A017437 A281264 * A366869 A267103 A030553
KEYWORD
nonn,easy,fini
AUTHOR
Jianing Song, Apr 23 2024
STATUS
approved