A267648 a(n) = g_n(5) where g is the function defined in A266202.

5, 9, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Author

Natan Arie Consigli, Mar 17 2016

Keywords

• fini
• nonn
• full

Comments

For more info see A266201-A266202.

This sequence has A266203(5)+1=62 terms and a(A266203(5))=a(61)=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(5)-1)/2)=a(30)=(A266203(5)+1)/2=31. - Zhuorui He, Aug 08 2025

Examples

			g_1(5) = b_2(5)-1 = b_2(2^2+1)-1 = 3^2+1-1 = 9;
g_2(5) = b_3(3^2)-1 = 4^2-1 = 15;
g_3(5) = b_4(3*4+3)-1 = 3*5+3-1 = 17;
g_4(5) = b_5(3*5 + 2)-1 = 3*6 + 2-1 = 19;
g_5(5) = b_6(3*6 + 1)-1 = 3*7+1-1 = 21;
g_6(5) = b_7(3*7)-1 = 3*8-1 = 23;
g_7(5) = b_8(2*8+7)-1 = 2*9+7-1 = 24;
g_8(5) = b_9(2*9+6)-1 = 2*10+6-1 = 25;
g_9(5) = b_10(2*10+5)-1 = 2*11+5-1 = 26;
g_10(5) = b_11(2*11+4)-1 = 2*12+4-1 = 27;
g_11(5) = b_12(2*12+3)-1 = 2*13+3-1 = 28;
g_12(5) = b_13(2*13+2)-1 = 2*14+2-1 = 29;
g_13(5) = b_14(2*14+1)-1 = 2*15+1-1 = 30;
g_14(5) = b_15(2*15)-1 = 2*16-1 = 31;
g_15(5) = b_16(16+15)-1 = 17+15-1 = 31;
...
g_30(5) = b_31(31)-1 = 31;
g_31(5) = b_32(31)-1 = 30;
g_32(5) = b_33(30)-1 = 29;
...
g_61(5) = 0. (End of sequence)
		

Crossrefs

Cf. A266204: G_n(5).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A266202: g_n(n); A266203: a(n) = k such that g_k(n)=0.

Programs

Mathematica

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 5], {n, 0, 61}] (* Michael De Vlieger, May 17 2016 *)

PARI

a(n, m=5) = { my(wn = m); for (k=2, n+1, wn = fromdigits(digits(wn, k), k+1) - 1); wn; }
vector(62, n, n--; a(n)) \\ Michel Marcus, Apr 03 2016 and Aug 08 2025

Extensions

Duplicated a(31) removed by Zhuorui He, Aug 07 2025