A266203 Number of steps k to make g_k(n) converge to zero.

0, 1, 3, 5, 21, 61, 381, 2045

Offset: 0

Author

Natan Arie Consigli, Jan 22 2016

Keywords

• nonn
• hard

Comments

Next term is 3*2^402653211 - 3;

g_k(n) is the weak Goodstein function defined in A266202.

For a complete table click the link below, and see table of upper bounds on weak Goodstein sequence.

Examples

			Find a(4):
g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;
g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;
g_3(4) = b_4(2*4 + 1 ) -1 = 2*5 + 1-1 = 10;
g_4(4) = b_5(2*5) -1 = 2*6 - 1 = 11;
g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;
g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;
g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;
g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;
g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;
g_10(4) = b_11(11)-1 = 12-1 = 11;
g_11(4) = b_12(11)-1 = 11-1 = 10;
g_12(4) = b_13(10)-1 = 10-1 = 9;
g_13(4) = b_14(9)-1 = 9-1 = 8;
...
g_21(4) = 0 so a(4)=21.
		

Links

• Googology Wiki, Weak Goodstein Table

Crossrefs

Cf. A056041, A266202.

Formula

a(n) = k such that g_k(n)=0.

a(n) = A056041(n)-2. - Pontus von Brömssen, Aug 08 2025