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

1,3

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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
(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<<L; n\=3^L); c}, [0..99]) \\ M. F. Hasler, Jul 13 2024

Crossrefs

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

Sequence in context: A197775 A002321 A043530 * A164995 A294108 A055718

Adjacent sequences: A297768 A297769 A297770 * A297772 A297773 A297774

Keyword

nonn,base,easy

Author

Clark Kimberling, Jan 26 2018

Status

approved