A297771 Number of distinct runs in base-3 digits of n.

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 2

Offset: 1

Author

Clark Kimberling, Jan 26 2018

Keywords

• nonn
• base
• easy

Comments

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences.

Examples

			1040 in base-3: 1,1,0,2,1,1,2; five runs, of which 3 are distinct, so that a(1040) = 3.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A043555 (number of runs, not necessarily distinct), A297770 (this for base 2).

Programs

Mathematica

b = 3; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]]
Table[s[n], {n, 1, 200}]

PARI

apply( {A297771(n)=my(r=Vec(0, 3), c); while(n, my(d=n%3, L=valuation(n+if(d>1, 1, d, n+1), 3)); !bittest(r[1+d], L) && c++ && r[1+d] += 1<M. F. Hasler, Jul 13 2024

Python

from itertools import groupby
from sympy.ntheory import digits
def A297771(n): return len(set(map(lambda x:tuple(x[1]),groupby(digits(n,3)[1:])))) # Chai Wah Wu, Jul 13 2024

Formula

a(n) << log n. In particular, a(n) <= log(n)/log(3) + 1. - Charles R Greathouse IV, Jul 13 2024