A297783 -id:A297783 - cp's OEIS frontend

A297770 Number of distinct runs in base-2 digits of n.

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

Offset: 1

Clark Kimberling, Jan 26 2018

Every positive integers occurs infinitely many times.
***
Guide to related sequences:
Base b      # runs    # distinct runs
      A005811       A297770
      A043555       A297771
      A043556       A297772
      A043557       A297773
      A043558       A297774
      A043559       A297775
      A043560       A297776
      A043561       A297777
      A043562       A297778
      A043563       A297779
      A043564       A297780
      A043565       A297781
      A043566       A297782
      A043567       A297783
      A043568       A297784

			27 in base-2: 1,1,0,1,1; three runs, of which 2 are distinct:  0 and 11, so that a(27) = 2.

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

Cf. A005811 (number of runs, not necessarily distinct).

b = 2; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]]
Table[s[n], {n, 1, 200}]

apply( {A297770(n)=my(r=[0,0], c); while(n, my(d=bitand(n,1), L=valuation(n+d, 2)); !bittest(r[1+d], L) && c++ && r[1+d] += 1<>=L); c}, [0..99]) \\ M. F. Hasler, Jul 13 2024

a(n) = my(s=strjoin(binary(n)), v=vecsort(concat(strsplit(s, "1"), strsplit(s, "0")), , 8)); #v-(v[1]==""); \\ Ruud H.G. van Tol, Aug 05 2024

from itertools import groupby
def A297770(n): return len(set(map(lambda x:tuple(x[1]),groupby(bin(n)[2:])))) # Chai Wah Wu, Jul 13 2024

A043567 Number of runs in base-15 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Clark Kimberling

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

			For n = 226, its base-15 representation is "101" as 226 = 1*(15^2) + 0*(15^1) + 1*(15^0). "101" has three runs, thus a(226) = 3.
For n = 482, its base-15 representation is "222" as 482 = 2*(15^2) + 2*(15^1) + 2*(15^0). "222" has just one run, thus a(482) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A043289, A043542, A297783 (number of distinct runs), A297770.

Table[Length@ Split@ IntegerDigits[n, 15], {n, 0, 105}] (* Michael De Vlieger, Oct 10 2017 *)

(define (A043567 n) (let loop ((n n) (runs 1) (pd (modulo n 15))) (if (zero? n) runs (let ((d (modulo n 15))) (loop (/ (- n d) 15) (+ runs (if (not (= d pd)) 1 0)) d))))) ;; Antti Karttunen, Oct 10 2017

More terms from Antti Karttunen, Oct 10 2017

Updated by Clark Kimberling, Feb 04 2018

A043568 Number of runs in base-16 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2

Offset: 0

Clark Kimberling

Differs from A043543 if we consider for example numbers which are palindromic in base 16 with 3 (2 distinct) digits. - R. J. Mathar, Oct 20 2008

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. - Clark Kimberling, Feb 04 2018

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A297783 (number of distinct runs), A297770.

b = 16; s[n_] := Length[Split[IntegerDigits[n, b]]];
Table[s[n], {n, 0, 200}]

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A297770 Number of distinct runs in base-2 digits of n.

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A043567 Number of runs in base-15 representation of n.

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A043568 Number of runs in base-16 representation of n.

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