A057650 Second step in Goodstein sequences, i.e., g(4) if g(2)=n: (first step) write g(2)=n in hereditary representation base 2, bump to base 3, then subtract 1 to produce g(3)=A056004(n), then (second step) write g(3) in hereditary representation base 3, bump to base 4, then subtract 1 to produce g(4).

1, 3, 41, 255, 257, 259, 553, 1023, 1025, 1027, 1065, 1279, 1281, 1283, 50973998591214355139406377, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084095

Offset: 2

Author

Henry Bottomley, Oct 13 2000

Keywords

• nonn

Examples

			a(12)=1065 since with g(2) = 12 = 2^(2+1) + 2^2, we get g(3) = 3^(3+1) + 3^3 - 1 = 107 = 3^(3+1) + 2*3^2 + 2*3 + 2 and g(4) = 4^(4+1) + 2*4^2 + 2*4 + 2 - 1 = 1065.
a(17) = 4^(4^4) - 1, with g(2) = 17 = 2^(2^2) + 1 and g(3) = 3^(3^3).
Similarly a(18) = 4^(4^4) + 1, with g(2) = 18 = 2^(2^2) + 2 and g(3) = 3^(3^3) + 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..1000
• R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
• Eric Weisstein's World of Mathematics, Hereditary Representation.
• Eric Weisstein's World of Mathematics, Goodstein Sequence.
• Wikipedia, Goodstein's Theorem
• Reinhard Zumkeller, Haskell programs for Goodstein sequences

Crossrefs

Cf. A056004, A059933, A059934, A059935, A059936.

Programs

Haskell

-- See Link