A237671 Let m_n denote the number which is obtained from n-base representation of m if its digits are written in nondecreasing order; then a(n) is the smallest period of the sequence which is defined by the recurrence b(0)=0, b(1)=1, b(k)=(b(k-1) + b(k-2))_n, for k>=2, or a(n)=0, if there is no such period.

1, 3, 16, 6, 20, 24, 16, 36, 120, 300, 20, 288, 28, 192, 200, 552, 180, 192, 180, 1380, 224, 60, 1728, 912, 3800, 756, 576, 1776, 4102, 15480, 3540, 1344, 10800, 14328, 800, 2304, 1520, 1890, 1232, 11280, 9040, 31152, 49544, 3660, 6360, 3696, 13248, 21408

Offset: 2

Author

Vladimir Shevelev and Peter J. C. Moses, Feb 11 2014

Keywords

• nonn
• base

Comments

We conjecture that the sequence b is always eventually periodic, and so a(n)>0.

Examples

			For n=5, b-sequence begins 0,1,1,2,3,1,4,1,1,2,... It has period {1,1,2,3,1,4} of length 6. So a(5)=6.
a(10) = 120, because the eventual period of A069638 is 120.
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 2..250 (terms 2..100 from Giovanni Resta)

Crossrefs

Cf. A069638, A237568, A004185, A306773.

Programs

Python

import sympy,functools
def digits2int(x,b):
  return functools.reduce(lambda n,d:b*n+d,x,0)
def A237671(n):
  return next(sympy.cycle_length(lambda x:(x[1],digits2int(sorted(sympy.ntheory.factor_.digits(sum(x),n)[1:]),n)),(0,1)))[0] # Pontus von Brömssen, Aug 28 2020