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

Author

Reinhard Zumkeller, Feb 13 2013

Keywords

• nonn

Comments

See A056004 for an alternate version.

Examples

			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.
		

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

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).

Programs

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