A222112 Initial step in Goodstein sequences: write n-1 in hereditary binary representation, then bump to base 3.

0, 1, 3, 4, 27, 28, 30, 31, 81, 82, 84, 85, 108, 109, 111, 112, 7625597484987, 7625597484988, 7625597484990, 7625597484991, 7625597485014, 7625597485015, 7625597485017, 7625597485018, 7625597485068, 7625597485069, 7625597485071, 7625597485072, 7625597485095

Offset

1,3

Comments

See A056004 for an alternate version.

References

Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

Example

n = 19: 19 - 1 = 18 = 2^4 + 2^1 = 2^2^2 + 2^1
-> a(19) = 3^3^3 + 3^1 = 7625597484990;
n = 20: 20 - 1 = 19 = 2^4 + 2^1 + 2^0 = 2^2^2 + 2^1 + 2^0
-> a(20) = 3^3^3 + 3^1 + 3^0 = 7625597484991;
n = 21: 21 - 1 = 20 = 2^4 + 2^2 = 2^2^2 + 2^2
-> a(21) = 3^3^3 + 3^3 = 7625597485014.

Prog

(Haskell) -- See Link
(PARI) A222112(n)=sum(i=1, #n=binary(n-1), if(n[i], 3^if(#n-i<2, #n-i, A222112(#n-i+1)))) \\ See A266201 for more general code. - M. F. Hasler, Feb 13 2017, edited Feb 19 2017

Crossrefs

Cf. A056004: G_1(n), A057650 G_2(n), A056041; A266201: G_n(n);

Cf. A215409: G_n(3), A056193: G_n(4), A266204: G_n(5), A266205: G_n(6), A222117: G_n(15), A059933: G_n(16), A211378: G_n(19).

Sequence in context: A151372 A258103 A300373 * A032832 A041021 A041022

Adjacent sequences: A222109 A222110 A222111 * A222113 A222114 A222115

Keyword

nonn

Author

Reinhard Zumkeller, Feb 13 2013

Status

approved