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

0,11

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A043536 A043561 A297778 * A043537 A047726 A297779

Adjacent sequences: A043559 A043560 A043561 * A043563 A043564 A043565

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved