A043562 Number of runs in base-10 representation of n.

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Blecksmith, Filaseta, & Nicol show that lim a(k^n) = infinity whenever k is not a power of 10. More generally, in base b, the limit is infinity exactly when log k/log b is irrational. - Charles R Greathouse IV, Jan 29 2014

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

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10000
• Richard Blecksmith, Michael Filaseta, and Charles Nicol, A result on the digits of a^n, Acta Arithmetica 64 (1993), pp. 331-339.

Crossrefs

Cf. A297778 (number of distinct runs), A297770.

Programs

Mathematica

Table[Length[Split[IntegerDigits[n]]],{n,0,90}] (* Harvey P. Dale, Aug 24 2016 *)

PARI

a(n)=my(d=digits(n)); #d-sum(i=2,#d,d[i]==d[i-1]) \\ Charles R Greathouse IV, Jan 29 2014

Python

def a(n): return len(s:=str(n))-sum(1 for i in range(1, len(s)) if s[i-1] == s[i])
print([a(n) for n in range(90)]) # Michael S. Branicky, Jan 08 2025 after Charles R Greathouse IV